Derivatives

A. M. Bruckner; J. L. Leonard

The American Mathematical Monthly, Vol. 73, No. 4, Part 2: Papers in Analysis. (Apr., 1966), pp.24-56.

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DERIVATIVES

A. 31. B R I J C K N E R AKD J . L. L E O K A R D , IJliiversiry o f California. Sailta Barbara

1. Introduction. In recent \-ears there has been a considerable aii io~int of research devoted to cl~iestions involving the derivative of a f~inction of one real variable and its generalizations. This activity is due, in par t , to the fundaniental role pla?-ed by the derivative in niathenlatics, and , also, to the difficulty of sorile of the interesting unsolved probleiils related to derivatives. I t seems appropriate tha t sollie of the results of this activit)., along \\-it11 sonie of the interesting bu t lot-so-n-ell-k~lo\\-n earlier res~ilts, be brought together and exanlined in one place. This is one of the purposes of the present expository article.

In deciding \\-hich topics to include in this article, \ye have given preference to ones which can be discussed \\-ithout first having to develop a great deal of machiner).. In addition, 11.e have leaned to~vard topics in I\-hich recent \york has been done and for I\-hich unsolvecl problems can be stated.

Froiii the long list of references given a t the end of this article, n-e single out the reference [ 1 9 2 ] .:\Ian>- of the recent n-orlcs on derivatives have their origin in this penetrating stud\- by Zahorslii.

2. Preliminaries. In this section \ye present a fen- of the definitions and con- cepts \\-hich appear in the sequel. T o avoid having this discussion beconle pro- hibitively long, \ye restrict ourselves to those notions n.hich appear pron-iinentl>- later on. For other real variable concepts n-llicli appear in this article (for ex- ample: approxii~late continuit)., F , sets, big and little o notation, densit\- condi- tions) the reader is referred to the texts [ 5 0 , 5 1 , 5 6 , 6 3 , 6 4 , 7 1 , 132 , 133 , 1 7 3 , 1 8 0 1 .

Througl~out this article \ye shall be concerned \\-it11 real valued functions of a real variable, usuall\- delined on an interval [a,b ] . Sucll a function belongs to Baire class 1 if it is the limit of a seclLieiice of con t in~ io~is f~inctions. \Ye define the other Baire classes inductivel!-: if oc is a countable ordinal, then ,I' is in Baire class cc provided it is the liiilit of a sequence of functions each of \\-hich is in a Baire class whose index is less than oc. Detailed studies of the Raise classes can be found in [ 3 , 5 1 , 5 6 , 8 3 , 133 , 1801. 17-e note tha t our definition is 1x11 incli~sive one: if J is in Baire class /3 and oc>p then,/ 'is also in Baire class a. In sonle s t ~ ~ d i e s it is niore convenient to have the Baire classes p;lir\\-ise disjoint. This is the case, for example, in the recent text 11331. I t is clear tha t ever\- derivative is a function in Baire class 1. . X propert\- possessed by every derivative of a continuous function is the

Darboux propert).. T h e function ,f satisfies the Uarboux condition (or j' is a 1)arboux function) on [a,b ] provided the set f [ I ] is connected for every interval I C [a,b ] . This propert\- is often called the i l z t e~ lncd ia t e.~la!i~epiopcrt?l, because a function has the Darboux property if and only if \\-henever XI and x2 are points of [a, B ] and 31is a nuiiiber bet~yeen ,/(.x1) and j'(x2), there is an ,\.a betn-een sl and .xq such tha t f (x j ) =y.

DERIVATIVES 25

Although there are many articles ill research journals ~vhich deal nit11 Darboux functions, a systematic study of such functions has escaped the standard real variable texts. I t is n o r t h noting tha t the Darboux property is far weaker than the property of continuity. Thus, there exist functions 11hich take on every real value on every perfect set [60]. Such a function obviously satisfies the Darboux condition bu t is non here continuous. Another indication of the size of the class of Darboux functions is the fact tha t ever)) function is the liinit of a sequence of Darboux functions [37]. One more reinark: the definition requires tha t the image (not the graph!) of a connected set be connected. There are Darboux-Baire functions whose graphs are not connected [84: p. 821. H o ~ ~ e v e r , it is shon n in [85] tha t i f f is in Baire class 1 , then f satisfies the Darboux condi- tion if and only if the graph off is connected. (See also [24; 62: pp. 289, 290; 84 : p. 811.) ( I t follows t h a t ever)- derivative has a connected graph.) For further remarks on this subject see [107]. T h e reader interested in Darboux functions is referred to the survey article [13].

\Ye end this section by stating for reference the definition of two rather com- plicated density conditions due to Zahorski [192]. \Ye refer to these iinportant conditions in Sections 4, 12, and 14.

DEFINITIOK.A nonempty set EC [a, b ] i s said to be a n 11f4set provided E is o! type I;, and there exists a sequence { I;,,)of closed sets and a seqzlence ( r , , )of nzim- bers, 0 <r,, < 1, such that E = U,:, F,, and for each x E F,,and every c> G there is a number ~ ( x , C)> O enjoying the following property: for any numbers h and hl szich that hhl>O, h / h ~ < c , I h+hll <e(x, c) we have

'\Ye note tha t in this definition I\-e require tha t the numbers r,,can be chosen to be strictl3- positive. If we relax this requirement to allon ing some or all of the r,, to be zero, we arrive a t the definition of an M3set. T h e definition of 111)set can be stated in a different and perhaps simpler form: the set E is an -113set provided t h a t if x E E and {I,)is a sequence of intervals not containing x such tha t {I,,) for all n , then --tx and m(I , ,AE) = O

(The condition ,114 cannot be given in an analogous manner, see Lipilislii [91].)

3. Continuity of the derivative. T h e student who has completed a first course in calculus is often not aTvare of the fact t h a t the derivative of a differentiable function need not be continuous. In a later course he learns t h a t the function j 1

given by fl(x) = x 2 sin ( l / x ) , fl(0) = O is differentiable, b u t fl fails to be con- tinuous a t the origin. He inight never learn, hon-ever, just how badly discon-

2 6 PAPERS I N ANALYSIS

tinuous a derivative can be. In this section TI e consider some questions concern- ing the continuity of derivatives. Our discussion points out , in addition, some of the pathological behavior possible of a derivative.

T o she\\ t h a t not ever) bounded derivative is Riemann integrable, I'olterra [182] gave an example of a function f z ivhose derivative is bounded bu t discon- tinuous on a set of positive (Lebesgue) measure. T o construct such a function, he considered a nolvhere dense perfect set PC [o, 11 of positive measure, and constructed a function ~ v l ~ i c h on each interval contiguous to P behaves, roughly, as the function fl (above) behaves on [O, I ] . This function, f 2 , was pu t together in such a way as to be differentiable on [0, 11 and to produce on all of P the singularity fl exhibits a t the origin. Llore precisely, f; =0 on P, but f; oscillates betvieen -1 and 1 in every neighborhood of an arbitrary point of P. I t follows tha t f; must be discontinuous on P. For a precise formulation of such a function see Goffman [51: p. 2101, Hobson [63. pp. 490, 4911, or Thielrnan [173 : p. 1651. A construction of the t3-pe referred to is possible relative to any non here dense perfect subset P of an interval I . Such subsets can have measure arbitraril3-close to the measure of I, b u t since I-P contains a dense open set, the set P cannot have full measure. I t is natural to ask just how large the set of discon- tinuities of a derivative can be. Does there exist, for example, a derivative nhich is everywhere discontinuous? T o see tha t this question must be ansn-ered in the negative, vie need only observe tha t a derivative f' is of Baire class 1 from n-hich it follox\-s tha t f ' must be continuous on a dense set [133: p. 1431. iVe neaken our requirement: is i t possible for a derivative to be discontinuous except on a de- numerable set? Again the ans\irer is "no." T o see this lve recall first t h a t the set of points of continuity of a n y function must be a Gs.Since this set must also be dense, as was seen above, it cannot be denumerable, for a dense Gs must be non- denumerable. (This fact folloxvs readily from the Baire categor) theorem.)

\Ye next ask TT hether or not i t is possible for a derivative to be discontinuous on a dense set. T o see tha t this question has an affirmative answer, xve use the follon ing approach. IT7e seek a differentiable function ivhose derivative vanishes on one dense set b u t is different from zero on another. Such a derivative must, of course, be discontinuous a t every point a t which it does not vanish. The problem of constructing such derivatives is quite old. In 1887 Kopcke [77] claimed to have given an example of a function f possessing the folloiving prop- erties. (a)f has a bounded derivative f1on [0, 1 ] ; (b) the set on \\ hich f' is posi- tive is dense in [0, I ] ; and (c) the set on xvhich f1 is negative is also dense in [o, I ] . Kopcke's original paper had a flaw which he corrected subsequently [78, 791. There follo\ved a sequence of articles on the subject, culminating in 1915, with a lengthy and penetrating study by Denjoy [31]. In this stud) the author provided a detailed discussion of differentiable functions whose derivatives take on both signs in ever) interval. I-Ie comments on some of the previous norlis on the subject, including Kopcke's, and gives several methods of constructing such functions. Zalcviasser [196] investigated the relative maxima and minima of such functions, obtaining results such as the following: L e t A a n d B be arb i t rary

nonoaer lnpp ing d e n ~ ~ m e r a b l e s i ~ b s e t so j [ a , 111. Tl zen tlzere emis fs a d i . f e r e n t i a b l e j u n c - f i o n f , h a v i n g a bounded der icat ive , such t ha t A i s the set of strict local m a x i m a o j f a n d R i s tlzc set of strict local m i n i m a o f j .

of the Kopcke t>-pe are too complicated to discuss here. I-lo\\-ever, a related kind of function, the so-called function of l'onipeiu 11511, 1s easier to '

understand and still exhibits the propert>- t h a t its derivative vanishes on one dense set b u t is different froin zero on another. A\gain, the derivative of such a f u n c t i o ~ ~is disconti~luous a t every point a t I\-hich it does not vanish. L,et us ob- serve t h a t if d is any real number then the function (.t. -d)1'3 has a finite deriva- tive except a t d , a t I\-hich point the derivative is infinite. L,et ( A , , ) be an>-sequence of positive umbers such t h a t ZA,,< m , and let i d , ) be an>- d e n ~ i n ~ e r - able dense subset of [0, 11. Then the series ZA,,(x-d,,)l:"efines a strictly in- creasing function f.I t can be shon-n t h a t f has a finite positive derivative a t all points for ~vhich the difierentiated series C + ~ , , ( . z . - d , , ) - " ~converges, and an infinite derivative other\\-ise. I t can further be shoxvn t h a t the inverse function

f p l is a strictly increasing differentiable function and its derivative vanishes on a dense set. ;I lengthy and detailed s tudy of such functions can be found in 31arcus [ 1 0 6 , 1131 . See also [ I 11 and [ 9 0 ] for answers to sorile cluestions raised in [ 1 1 3 ] .

Diflerentiable functions \\-hose derivatives are discontinuous on a preassigned den~imerable set i d , , ) (n-hich ma>- be dense) can be constructed by considering any uniformly convergent series of derivatives, X i , , , such t h a t the function f,, is discontinuous only a t d,,. 1-or exanlple, the function j given b>-f ( ? c ) cos ( x - a , , ) - l is a derivative x i t h discontinuities on the set = x:==, ( d , ,) . (See Ilalperin [611].)

l y e have seen t h a t the set of discontinuities of a derivative can be dense, bu t t h a t the set of points of continuity niust also be dense and must be nonde- numerable. Finally, J\-e ask: what are necessary and sufficient conditions 011 a set E t h a t it be the set of discontinuities of a derivative? I t is easy to verify t h a t such a set 111ust be an F, of the first category. C:onversely, if E is a n y first category F,, E C [ a , b], then E can be expressed as the u n i o ~ ~ of an expanding sequence of nowhere dense closed sets E,,. 14-it11 each E,, TI-e associate a "1701terra type" function f, \vith the propert>- t h a t if x c i E E , ,then f?:oscillates hetn-een -1 and 1 in each ~leighhorhood of -2,). I t is not hard to verify tha t the f u n c t i o ~ ~ /' defined by J ( x ) = C f , , ( x )i31Lis differentiable on [ a , b ] and its derivative is con- tinuous a t each point of WE,b u t d i s c o n t i ~ l ~ o ~ s a t each point of E. Thus \\-e have

TEIEOREAI.A zeces sary a n d s l~ f l i c i en t c o n d i t i o n t ha t a set EC [ a , b ] be the set o j d iscont inz t i t ies of" a derielative, i s t ha t E be a n F, of the ,first ca tegory . (-Although lye-imagine t h a t this theorem is knon-11, we have bee11 unable to find a reference.)

111 particular, there are d e r i ~ a t i ~ e s a.e. 01111hich are d i s c o n t i n ~ ~ o ~ i s [ a , b ] . For if to each positive integer n n e 111;llte correspoild a 11011here dense closed sub- set E,, of [ a , b ] having nleasure greater than b - a -1l?z, the11 the set E \\ hich is the union of the E,,'s is a first c a t e g o r ~ F, of me;lsure b-a. T h e result follons fro111 the theorem above.

4. An unsolved problem. -\1311> classes of fu~lctions can be characterized 111

ternis of 11hat the inverse lapping of a function in the class does to cert,iin open sets 'The chart belo\\ suniniari~es some of these c11,~racteri~ations. Let a be a11y real number and let

7 hen f i s if u n d o n l y 1Jfoi (111 reczl a , @

co~ititiuous E,(f) and E a ( j )are ope11 Raire class 1 E,(j) and E a ( j )are sets of type F, Raire class E ( E n cou~itable ord i~ la l ) E a ( J )and E a ( f )are additive Borel class [ if E finite. :+I if E

infinite iri sorne Bnire class E,(f) and E a ( j )are Borel sets upper semi-cont inuo~~s E a ( j )is open lower s e m i - c o ~ l t i ~ ~ ~ ~ o ~ ~ s E a ( j )is open measurable E U ( J )atid E a ( f )are iileasurable approximatel\- contiriuous each . v c E a ( J ) n E p ( f )is a point of density of t h a t set, and

t h a t set is all F,.

I t is ~ la tura l to ask what the correspoildi~lg characterizations are for various classes of derivatives. This question has bee11 studied by Zahorslii [192].111 this I\-ork, he found necessarJ- collditioils and also sufficient coilditio~ls in ter~l ls of the sets Em(!) and Ea(j')for a fu11ctio11 to be a bou~lded derivative, a finite derivative, or a derivative, possibly infinite, bu t he \\-as unable to find character-i z a t i o n s of these classes of derivatives. (See Section 14 for a illore detailed discus- sion of Zahorski's results.) 'The q u e s t i o ~ ~ of characterizatioil of these classes is still open.

5. Derivatives a.e. and universal generalized antiderivatives. As \Ye san-in Section 4, the problei~i of characterizi~lg derivatives in terms of the sets E a ( f )and/or E,(f) has not yet been resolved. \Ye turn low to the problem of fi~ldi~lg; ~vhich have the prop- such a characterizatioil for the class of fu~lc t io~ls ertj- of being; a l m o s t eoerywhere the derivative of a continuous f u n c t i o ~ ~ . As n e shall see, this requ i re~ue~l t i~nposesvery little restriction on a fu~lction.

14-e first observe tha t every Lebesgue summable fu11ctio11 is al~llost every- nhere the derivative of its integral. The same is true of any fu11ctio11 integrable in the sense of Lknjoy-Perron. (See Section 10.) There are, ho\\-ever, ~lleasurable functio~lsnot integrable in either of the above senses. 111 1915, Lusin [97, 991 published the follo~villg theorenl, \vhich corupletely solves the proble~n of char-acterizing those f u ~ ~ c t i o ~ l s \\-hich are derivatives a.e. of co~l t i~ luousfunctio~ls: E o e r y m e a s u r a b l e f u n c t i o n f ( f ini te a .c . ) i s a l m o s t everywhere tile der ivat ive of a c o n t i n u o u s J u n c t i o n F.The conditio~l of measurability of f as \\-ell as tha t of a.e. finiteness is obvious1~- necessary as \\-ell as sufficient. Thus, the a .e . jiiinite Jzrnction f i s a.e. the der ivat ive of a cont inz tous Jz tnc t ion I: iJ a n d o n l y i f f i s nzea- .s~trnble, or eqzrielalently, iJ a n d o n l y i f each set of the fornz {x:f(m) < a { or of the fornz i z : f(m) > a ) is measl t rable . Of course the fu~lctioil f is not the only one

DERIVATIVES 29

11-hich is a .e , the derivative of F, b u t any other such function is equivalent to j (i.e., agrees n-ith f a.e.).

Let us see what happens if IT e ~veaken the requirement of being a.e. a deriva- tive still further. Let f be an arbitrary function on [a, b ] and suppose there exists a sequence { h,, 1 of numbers n ith h, J 0 and a continuous function E' such tha t

F(x + h,,) - F(x)j(x) = lim n-. rn h n

a.e. on I=[a, b]. Then F may be called a generalized antiderioatioe of f . I t is clear t h a t such an F may be a generalized antiderivative of many fu~lctions not equivalent to f . How many? AIarcinlcie~vicz [101] has proved the follo~ving remarkable theore~n: There exists a continzlozls function I.' which is a generalized antiderivatioe for eoery use. jinite measzlrable function. (Tha t is, F is a universal generalized antiderivative.) I t Ivas also sho~vn in [101] tha t most functions are universal generalized antiderivatives, in tha t the class of colltinuous fullctio~ls which are not universal generalized antiderivatives form a set of the first cate- gory in ~ [ a , b ] . A proof of the theorem of Lusin and mention of the theorem of IIarcinkieTvicz can be found in [157: pp. 215-2183.

Other results of the above type have been obtained bj- Sierpihslci [163] and Eilenberg and Saks [35]. One such result [35] deals ~ v i t h the generalized anti- derivative of arbitrary (not necessarily nleasurable) functions: Let f be any fzlnc- tion defined on an interoal I and let H be any denzlmerable set of real nzlmbers. Then there exists a continzlous function F such that

f (x) = lim F[(x 4- lz,) - ~ ( x ) ] / h , n 3 m

for every null sequence (h,,{ from H. (There is no exceptional set here; the result holds for all x.)

6. Dini Derivatives. *A fu~lction defillecl oil an interval I= [a, b ] has defi~led a t each point of I four Dini derivatives (except a t a and b , a t ~vhich onll- t ~ v o of the Dini derivatives are defined). For example, the upper right Dini derivative of f , D+f , is defined bl-

f (X+ h) - f(x)D+f(x) = lim sup - - s

and the other three are defined in an analogous manner. An elementary result is t h a t if one of the Dini derivatives of f is continuous a t a point xO,then j is differentiable a t xo.

In 1915 Denjoy [30] proved a theorem relating the four Dini derivatives for continuous functions. This \\as generalized to measurable functions bjr Young [189] in 1916 and to arbitrary functions by Salcs [156] in 1924.

30 PAPERS I N ANALYSIS

T I - I E O K E ~ I .Let f the possible exceptio?~ bejinite o?z [a, 1 1 1 . Then, ~ ~ ' i t l l o/' a nu?? set, [a , b ] can be decomposed i?ztojoztr sets:

Al , on which f 1las a jinite (ordinary) deritlatiue, A 2 , o?z wllicll D+f= D-f (j inite), D-f = a,D+f= - a,

A3, on wllicll D+j'= a , D-f = - a , D-j'= D+f (ji?zite), and Ad,on which Dij'= D-f = m , D+f=D-f= - 00.

T h e theorem is valid if one replaces [a, b ] by any set d (not ~lecessaril?; nlea- surable).

So~l leinlnlediate coIlsequeIlces of this theorem are the follon-ing : (1) aA1l increasing function is differentiable a.e. (for the sets A" ,A3, and A 4

are empty i11 this case) ;

( 2 ) fuilctio~l of bounded variation is differentiable a.e. (for such a func-tion is the difference of t~x-o increasing fu~lct io~ls) ;

(3) If f is finite 011 [a , h ] , then the set 011 ~vllichf' is infinite is a 111111 set. ( I t is interesting to observe, by IT-a?; of contrast , t h a t there exist functions n-hich have D+J= rn , even though /' is r ight-co~lti~luous. c a ~ l n o tHere, right-co1lti11uit~- be replaced by continuity. See [7: pp. 125, 126; 124; 1691.)

T h e Denjoy-You~lg-Salts theore111 has bee11 extended by Garg [ 4 2 ] ,1~110

shoived t h a t the exceptio~lal 11~111set also has ail image of measure zero. Garg has also co~lsidered the set a t ~vhich the L3i1li derivatives vanish. Some applicatio~ls of this exte~ls io~l derivatives of may be found i11 [43,44, 451. For results on D i ~ l i IIOIT-heremonoto~lefu~lctions,the reader is referred to Garg [46, 47, 481.

For continuous functions the sets { Dif# D-f ] a~l t l { D+f z D-/'} are sniall ~ I Ithe sense of category. T h e follo~\-i11g result is due to Seugebauer [140]: I f f i s co?zti?zltolts the72 tile sets (Di/'# D-/'] and { D+J# D - f } are of.rirst category. I f in additiox J i s o/' boz~?zded oariatio?~ on eoery closed intertlal, thez these sets are of measlire zero as well. The characteristic fu~lction of the ratio~lals sho~vst h a t con- tinuity ca111lot be dropped fro111 the first statelnent; nor can the h~-pothesis of bou~lded var ia t io~l be dropped fro111 the seco~ld s ta teme~l t , as is sho11-11 b ~ - Ex-ample I11 of Ue~l joy [ S O ] . Seugebauer 's theore~u , as \\-ell as certain related re- sults, is a consequence of a result found ~ I I[203].

AAlthougha Dini derivative does not , in general, satisfJ- the Darboux condi- tio11, s o ~ u e interesting results about the intermediate values take11 011by Dini derivatives have been advanced b?; JIorse [130].One such result is the follo~\-- ing: I f f is conti?zuo~is, - = <X < a,if the set { x : D+f(x) ? A } is dense a d tile set ( x : D+f(x) < A ) is nonenzpty, tlle?z the set ( x : D+f(x) = A ) has the power of tile con f inz~um.

i4derivative (finite or infinite) of a real valuetl function is aln-ays in Baire class 1. 'The correspolldillg statement for Dini derivatives is not valid, even for co~ltilluous functions. F-lo\\-ever, if J is in Baire class a, then the four Dini derivatives are in Baire class a f 2 [162]and if f is measurable, then so are its Dini derivatives [ 4 ] .If f is not measurable, then the same may be true of its Llini derivatives. HOIT-ever, F-ILjek [58]has advanced the surprising result t h a t for any finite f u ~ l c t i o ~ l (nleasurable or not) the extreme Bilateral derivatives

DERIVATIVES 31

a nu st be of Bnire class 2. (The upper b i l~ t e r a l derivative of a function f is defined by

f(x + il) - f(x)J(x) = lirn sup .

?I -0 h

The lo~ver bilateral derivative is defined a~lalogousl>-.) 'This result is the best possible, for there exists a function j' satisfying a Lipschitz co~ltlition, I\-ith $7 not ill Baire class 1 [168].

7. Approximate derivatives. In certain ilistailces a function fails to have a derivative a t a point so ,).et the restriction of the fu~lctioll to a set nhose coniple- nlent is veq- "thin" near scihas a derivative a t s o . If one proper1~- interprets "tliir~"il l ternis of density, then one arri\-es a t the noti011 of an approximate derivative.

~ E ~ I Y I T I O Y . EL e t f br defi?zrd o?z [ a , b ] , a n d let s c i E ( a , b ) . lj thrvr rxzctc n ~ e t ~ Z I C J Z t ha t (1) s o E E , (2.) so i\ n 11olnt o f zevo dmzuzty w i t h r e i p ~ c t to -E, a n d (3)

e x i s t s , .for s vestuicted to E , t h e n t h i s l i m i t is cc~llcd the a,b,bvosiiizntc dcui.itntitlc 0-Fjc ~ t x o a n d i s zauitten Ji,(.xo). (The o l ~ v i o ~ ~ s made s o = a orinodificatio~ls are if x,i = b . )

The lotion of approxi~liate derivative 1~21s illtroduced by L1elljo)- [29] and plays an inlportant role in the theory of the Delljoj--l<hi~ltcl~i~le integral (see Section 10).

The approximate derivative also arises in co~l~ lec t io~l \\-it11 certain cluestio~ls illvolvillg the approximation of f ~ ~ n c t i o n s Th11s, let f beof several real varial~les. defined on, say, the unit square S ill Euclidea~l tn-o-space. According to L11sin's 'Theorem J is riieasurable if and 0111~- if for ever\- E >0 there exists a f~11lctio11g co~l t i~ luous011 S such tha t g=f except 011 solrle set of measure less than E .

Suppose II-e wish to approximate .f in this sense b>- a f~11lctio11 II-hich is not 0111)-continuous, I I L I ~also has a co~l t i~luous andtotal differe~ltial. This is possible if o111y i f f has an approximate total differential al~llost ever>-I\-here [216]. Tha t is, i11 order tha t f have the propert>- that for every E >0 there exists a c o ~ ~ t i ~ l u o u s l ~ -difiere~ltiable f ~ ~ ~ l c t i o l l g s11c11 tha t g =Jexcept oil a set of liieasure less t-llall E , it is necessar>- and sufficient that .f be approxi~natelj- differentiable a.e. 'This last co~ldi t io~lis equivalent to the co~ldi t io~l that the partial approximate deriva- tives o f f exist a.e. 12131, see also 115'9: p. 3001. (\Ye note that the corresponding s t a t e ~ n e ~ l t fu~lc t io~lfor ordi~lai-y partial derivatives is false. There is a co~ l t i~ l r~ous of ~ I I - o variables n-hose partial derivatives exist a.e. bu t n-hose total differential exists ~zozahcvc [213: p. 5151.)

,\pproximate derivatives possess some of the properties of ordinar>- deriva- tives. Thus, if f is approximatel>- tiifferelltiable on [ a , 1 1 1 , the11f:, is of Baire

32 PAPERS IS -%SALISIS

class 1 and possesses the Darboux propertq (see Ichintchine [73, 741 and Tolstoff [174]). For a unified develop~nent of these results, among others, the reader is referred to Goffinan and Seugebauer [53].

If one allons the approxinlate derivative to be infinite, then the situation is a little different. Zahorski [191] has given an example of a function j 11hich has a t every point an approxinlate derivative (finite or infinite), bu t fi, is not of Baire class 1, nor does it satisfq the Darboux condition. In the saine article, honever, he she\\-s tha t i f f has a t each point an approxiinate derivative (finite or infinite), then f:, is of Baire class 2. I n addition, f must be of Baire class 2 a s nell . In case f is approximately continuous and has a t each point a finite or infinite approxinlate derivative, then f:, (as n ell as f ) must be of Baire class 1. (See Tolstoff, 11741.) In addition, f,', nus st satisfl- the Darboux condition in this case 1821. Soine additional results involving the Baire class of approxiinate derivatives can be found in I<rzq zeu ski [81] and JIatl-siak [I191.

I t is of interest to note tha t nhile the set sf discontinuities of a fu~lction having everq nhere a derivative (possiblq infinite) must be denumerable, Lipirislti [92] has s h o ~ ~ n tha t the set of points of approximate discontinuitq of a function having ever)-\\ here an approximate derivative (possibll- infinite) can be nondenumerable, although the set must have zero measure aild be of the first category.

Under certain conditions a point of approxinlate differentiabilit) is actually a point of differentiabilitq . Thus if J is monotonic,j is differentiable \\-herever f is approximately differentiable [73, 741. Ichintchine has also shou n tha t if an approximate derivative (possibll- infinite) is dominated b> an ordinar)- deriva- tive, then this approxinlate derivative is in fact an ordinary derivative. This result has been used bq Tolstoff to prove t h a t i f f is approximately continuous and has a t each point a finite or infinite approximate derivativej;,, then except possibly on a no\\ here dense set, f:, is the ordinarq derivative of f. ( [175]; see also [53].)

111 1916 Denjol- [33] noted the f o l l o ~ ~ of '1 (finite) derivative: ing propertq If a </3 t h e n tize set EaB=[ x :a <f ' ( ~ ) zs eztizer e m p t y ov h a s povztiae rneasutpe. < / 3 } This result \ \as extended bq. Clarlcson [26] to derivatives (nhich might be infinite) of continuous functions. -4 more detailed description of the sets Eap\\.as advanced bq I-Isiang [65]. Finall!, in 1963, ;\Iarcus [lo81 s h o ~ ~ e d tha t the cor- responding results are valid if one replaces "derivative" bq ((approximate derivative" in the hj~pothesis , ~ n d conclusion of Clarltson's theorein. I n case one allons the approximate derivative to be infinite a t some points, additional as- sumptions are necessarl-, As nlentioned in Section 4, Zahorski [192] has obtained results concerning the structure of sets of the forin [ x , f l ( x ) < p } . Correspondiilg results for approxiinate derivatives and Peano derivatives have been advanced bq LYeil [185] and Iculbacka [82].

\T7hether or not a function is approxilnatelq differentiable, it aln aqTs has four extreme unilateral approxiinate derivates. For a detniled studq of these, the redder is referred to Jefferq [71]. Lye mention the interesting fact, 11hich ma>- be

fount1 011 p g e s 198-199 of [ ? I ] , that the theoreni of Dcnjo! -S,ll;s-170u~ig for Dini tieriv,ttives (see Sectio~l 6 above) has a v i r t l ~ a i l ~ide~ltical all,~log~ie. If f happe~lsto be measurable, the11 the sets I! hich correspo~lti to A?and A?(of Sec- tion 6) are 11~111sets. For results on ~lollliieasurnble fullctiolls, see Cho11 [25].

I t is of i~lterest to note that for fu~lctiolls of s e t l e ~ u lvariables the extren~e ulli-lateral partial approximate derivatives of a function ,f reflect the measurability properties of f , n-hereas the partial Dini derivatives do not. Thus if f is a Le- hesgue (Borel) l ~ ~ e a s ~ ~ r a b l e real valued fullctioll of several variables, the saiile is true of its extrelrle unilateral partial approxilliate derivatives. 011 the other hand, there are Lebesgue (Uorel) nleas~~rable functions of tn-o vnriables n-hose partial Llini derivatives are not Lel~esgue (resp. Borel) ~iieasurable. I t is true, ho\\-ever, that if j'is colltilluous (Bore1 ~lieasurahle), the11 its partial Di~i i cleriva-tives are Uorel measura1)le (resp. Lehesgue measural~le). (For fu~lctions of one real variable, the Dini tierivatives as n-ell as the extreme u~lilateral approxi~ilate derivatives inherit the Lebesgue or Borel nieasurability of the primitive func- tion.) For results of this sort see [83 : p. 3211, [157: pp. 113, 171, 2991, and 12091.

\Ye conclude b ~ - mentioning that the 11-ord "thin" nientioneti in the i~ltroduc- tor)- paragraph can be i~lterpretecl in other I\-a).s, giving rise to different sorts of derivatives. Thus, for exanlpIe, S.:\larcus [ill, 1121 has interpreted "thin" in terms of categor). (rather than nleasure) and arrived a t the notio~l of a clunlitn-tive derivative. The notion of "preponderant" derivative, due to Denjoy [29], is related to the notion of approxiinnte derivative but the complements need not be quite so "thin" for a prepolldera~lt derivative to exist as for an approxililate derivative to exist. For a fuller discussion of this matter in a slight1~- broader context, consult Section 8.

8. Other generalizations of the derivative. I t is scarce1~- surprisiiig tha t such a f u ~ ~ d a ~ ~ i e ~ i t ; ~ lconcept as the derivative has received gelleralizatioil in a nun~ber of different directions for various special purposes. l\lall). gelleralizatio~ls are arrived a t b)- n-ealreni11g the sense i11 I\-hich the liinit of the difference cluotiellt [f(x+Jz) -f (x)],']I is obtained, altho~igh other avenues of definition are soine- times used. Irs11all~- the existence of the generalized tierivative together \\-it11 solrle regularity co~lclitio~l ilnplies the existence of the ordinar\- tierivative, anti the restrictivelless of this regularity collditioll call be used as all intiex of the degree of gelleralizatioll obtaiiied. \There the ordinar!- tierivative exists, it is ecl~ial to the generalizeti tierivative. We shall give the definitions of various gelleralizatiolls a ~ i d discuss l~riefly some of the Inore in~portallt ailloilg then].

The Dini derivates, discussed i11 Sectio~l 6, represent the first ge~leralizatio~l of the ordinary tierivative, in that I\-e do not restrict olirselves to tlie liinit of the difference cluotiellt, ~ ~ h i c h limit niay fail to exist, but rather considcr the one-sitied limit inferior and limit superior. 111 this \\-a>- 11-e are assureti of the existence of these derivatives a t each t\\-o-sided liinit point of the clolnnin of the f~illctio11.\\'e neeti 0111~-1)e given the continl~ity of one Dini derivate a t a point to collcl~ide the existence of the ordillary derivative a t that point.

T h e approximate derivative, discussed i11 Section 7 , is a ~ l a t u r a l ge~lcraliza- tion of the ordi11ar)- tierivative i l l 1\-11ich the liniit of the ciiflere~lce quotiellt is take11i l l the metric se~lsc of the approximate limit. -As poilltccl ou t i11 Section 7 , all approxilnatc1~- co~l t i~ luous XI approxilli;~te derivative every- fl~llctioll r i t h \\-here i l l an interval [ a , B ] possesses an o rd i~ la ry derivative on a set of i~l tcrvals \\-hich is tiense i l l [ a , b ] (see [ 1 7 4 ] ) .

IYeakening the density req~iiremcnts for the existc~lce of a limit i l l the defini- tion of a p p r o x i ~ ~ ~ a t e derivative (see Section 7) to t7zc set E has i v e n v density gventer t h a n 1 ' 2 o n a l l s i~_tj icicntly siizall i.rzter;itnls ii7clildii7g 30, 11-e obtain the prepo.rzdcvant derivates and derivative of Denjo\- [ 2 9 ] .Replacillg iiietric colisiderations \\-it11 the co~lcept of category, one arrives a t the ( zPProx i i i z c~ t~( j i ~ a l i t a t i z ~ cc1eri~-ativc tiefined by S. !\[arcus [ l l l , 1121, \\-here the upper qualitative limit of f at sois defined as inf (31:{ x : j ' ( x )>y ) is first categor)- a t x o ) , tlie lower qualitative limit is similarly de!ined, and these lililiti~lg o p e r a l i o ~ ~ s are applied to the cliflerellce cluotient to yield approximate clualitative clerivntes, 11-hicli share Illany of the properties of the Dini clerivates.

I n taking the liniit of the differellce quotient, I\-e may restrict ol~rselves t o co~lsiderillg olll\- v a l ~ ~ e s belong t o a given set E , 11-11ich set has s,,of su+72 \~-llich a s a limit point. This \\-ill give us the deri;itc~ti.ilc~ E . \Ia11)- of 0.f ,/' vclati;it~ to t he s ~ t the theorems foulld i11 Salis [ 1 5 7 ] hold for this forlll of the derivative. Sii~ii lar i l l

concept is the c o n g r ~ l e n t deri.ilntiilc of Sinclalovsliii [164 , 165, 212 1 , ill 11-11ich the values of h used in forniing the cliffere~lce cluotient arc restricted to belong to 2 set Q,I\-hich has 0 as a linlit point, h u t \\here this cliffcrence q~ lo t i cn t is defined for every .YE[ a , B ] , all\-a\-s using tlie same set Q.T h e idea of passing to the l i ~ n i t I\-hile ~leglectillg values obtai~lecl on "~legligihle" sets h c l o ~ ~ g i ~ i g to a particular family has heen advanced 11)- Csriszrir [ 2 7 ] .

Cha11gi11g the form of the difference cluotient gives rise to man\- generaliza-tions of the derivative. T h e most common is the s \ -m~ne t r i c derivative (also called the Riemann tierivative), clefi~led h\-

This derivative has the virtue of not i~lvolvillg the behavior of f a t the point s itself. I t is witiel~. used in the theor>- of trigonometric series. (See, for example, [198 , 1991.) T h e existence of the s\-mnietric derivative a t all points of a set E implies the existence of the ordinar)- derivative a.e. i l l E [ 7 3 ] .'l'he sj-nlnletric tierivative is nice1)- arrived a t through cleco~nposilig the fuliction j a t .yo into its c i G and odd par ts :

&,(t) = %[f(mo + t ) +S(.~O- t ) ] ,

lbT0( t ) = 3[f(xo + t ) - j ( n . 0 - t ) ] .

T h e differe~ltiahility of the odd par t a t t =O is thell equivale~l t to the existence of the sl-minetric derivative, I\-hile the cliff erentiahilit). of the even par t is ecluiv-

d e n t to the property 01 rvroofllnr.\\ [197, 1391 (see Section 9). 'The higher Riemann derivatives, given b \

extend the s~ ~r~ ine t r i c deriva-deri\r,xti\ e [I$, 20, 72, 1991 'I'lie second I i i e ~ n a n ~ i tive is often c'tlled the Schn nrL d e r i v a t i ~ e .

The difference q~io t ien t ma^ Ije \,tried in other directlo~ls. 'The simplest ~ , i r i a n tis

'This definition \\.as considered 11~- T'eallo 11461, n-ho felt tha t it portra~-ecl tlie concept of the derivative used in the phi-sical sciences more close1~- than does the usual defirlitior~, s i~lcef* and co i~~c ides is all\-a)-s c o ~ ~ t i n u o u s , 11-ithj" \\-ilenever ,f' is continuous. This definition has been recently reconsidered 13)- Esser and Shisha [ 3 6 ] .Another variation is Sindalovslti!'~ derivative [166],

11here I$ is an 'irbitrdr) defined in a neighborhood of the origin, 11hichl u ~ ~ c t i o n , approaclies 0 TI it11 11. l lu rav 'ev [131] dealt \I it11 tlie (;;tte,iux derix a t i ~ e,

\I here f ir differentiable (in the ordinar\ sense) on [a , [ I ] , and n ( ~ )is an) botinded function defi~lecl on [ n , [ I ] . The 1i11r'ir frrllctio~l i n the cienonlin,ttor of the difference quotient 111,x)- be exchanged for nn nrbitrar) function g ( u ) , J-ielding the deriv t~tzve w i t h respect t o g given b).

The physic,il sciences, in particul'ir the r~ l iod>-~ l~~tn ics ,led Hose1 to define n ??zen?zd~vivafive[10].

S ~ r g e n t [IS$] extended this defirlitior~ to p,xrallel the U i ~ i i deriv,ltes, nnd ll~ircinltienicz. ~ n dZl-glnund 11031 extended his res~l l ts to n "sl~iooth Bore1 deriv,~tive,"

I11 ;L master1~- paper, l<hintchine 17'31 co~lsidered various candidates for ;I

"ge~leralized derivative," estahlishi~lg their properties and constructing examples to illustrate a hierarch\- of generalit>-. T h e s>-nimetric derixrative n-as discarded since its existence, save on a 111111 set , implies the existerlce of tile ordinar)- deriva- tive x.e. T h e Rorel derivative generalizes the SJ-mlnctric derivative and laclis this "i-la\\-,"bu t is itself generalized bj- tile approximate derivative. 'The approxi- mate derivative, holyever, is generalized b), In dc'ri~6i. ge'1z6i.ule of J', I\-hich is a fullctiorl J,/ , defined 0111~- x.e., such tha t for an\- E >0,

lf(r + 11) - f(.v) .T : I -------- f; (n.) 1> E 17?z I 11

tends to 0 \\-it1111. T h e existence off(:, a.e, 011an interval implies the existence of ,/'; on the interval, and J;,,=j'!; a iu~lct iona.e. , \\-bile Khintchi~le c o n s t r ~ ~ c t e d .f'

such thxt ,/'I:exists on [O, I ] , and J,:,,0111). 011a null set. Hal\-ever, /',/ bol\-s to la de'riv6c g6fz6rc~lisc'r, , f c i n-hicll is :LII\- function \\-it11 , defined even less u ~ l i q u e l ~ - , the propert>- t h a t , for some sequence ( / I , , ] decreasing to 0 , 71.e have [ f ( s+ i~ , , )- J ' (~) ] , . ' h , ,~ j ' r ; 'a .e. This treatment reflects the generalized antideriva- tive of Section 5. l<llilltchine closed b ~ . co~lstructinga continuous fu~lctiorl \l.hicll fails to have exren a d6riv6r g6?1,6vi~lis&r.

;\ completely different approach, \\-hich yields a genera1iz;~tion of ordinar\- derivatixres of order greater than one, is given b ~ - to apolyliolnial approximatio~l function. If J(.vo+il,) can he expressed as

thc11 the Ji's are referred to ns the ith l'eano derivatives [145] (referred to bj- rlenjoy [32] as differential coefficients, and sometiines called de la 1-allCe r 'ouss i~~derivatixres [181], although this latter tern1 is also used othern-ise 111991). T h e 12th I'eano derixratixrej,, al\\-a\,s ecluals the 12th ordi~lary derivative \\-hen the latter exists. Oliver [143] studied the exact 12th I'eano derivative, I\-hich is one \\.hich exists a t every point of an interxral, and sho\\.ed tha t it is of Baire class 1, enjoys the 1)arboux property and the L)enjo>- propert), of Ems,and coincides 11-ith the ordinnry nth derivative on a dense open set. ;\ side condition for the existence every\\-here of the nth or dinar^- derivative is tha t the 11th I'eano deriva- tive be bounded either above or belon-. If the 12th Kiei l ia~l~l derivative exists ever\-\\-here on a set B of positive measure, then the 72th I'eano derivative exists a.e. on E [103].

C;eneralization of the Peano derixrative leads to the LPderivative: if jELlJ, 1s @5 cc , i11 some neighborl~ood of so,and if a polynonlial

exists such tha t

thenf is said to be differentiable of order ~z a t xo in Lp,and f , is the it11 L' deriva-tive. This generalization n-as introduced b! Calderhn and Zygmu~ld [ZI , 221 because the property of differentiabilitl in L?]a t a point is preserved under vari- ous integral transformations, and \\-as applied to solving partial differential equations. T h e Lp derivative has been used recentlh- to establish the differentia- bility a.e. of functions [136, 138, 1691 (see Sectioil 9). \Yeiss [186] coilsidered the symmetric kth derivative in LJ',and generalized the result tha t the existence of the kth s? mmetric derivative implies the existence of the kth (Peano) deriva- tive a .e . Higher dinlensiolls are considered in [215].

T h e n th T a l l o r derivative also arises out of considerations of polynomial approximation. I t is defined [19, 201 as

where is the ordinar? kth derivative. Butzer [19] and Gorlich and Sessel [55] studied the relationships betneen the Riemann, Peano, T a l l o r , and ordi- narq nth derivatives (listed in order of decreasing generality), \\-here convergence \\-as considered in both the usual metric and in the Lll nornl.

For those interested in further generalizations of the derivative, 11e men- tion the fluents [126] and multiderivative [125, 1271 of l Ienger , the I-Iolder and Cesaro derivatives of order n [152], the fractiollal derivative of Icuttner [86], Alinetti's right oscillatory derivative [128], O'Neill's vorlc on general i~ed derivates [144], and Shultla's on nonsq mmetric differentiabilit? [160, 161 1.

T h e extension of the concept of derivative to spaces other than the real line gives rise to a vast literature, ~ ~ h i c h we 11ill not explore. Iiltroductiolls to this field are provided by Bogel's paper on higher-dimensional differentiation [9] , FrCchet's s tudy [39] of various definitions of differentiabilit? in the plane, and the article of Rinehart and [Yilson on differentiation in algebras [153].

9. Points of differentiability. Since the existence of the derivative of a func- tion f a t a point iinplies a certain degree of good behavior off a t tha t point, i t is most natural to s tudy the set of such points of differentiability. I t is especially desirable to find under n ha t conditions this set is large, and i t is also of interest to Itno\\- something about the size of the image of this set under j.11-e discuss these questions in this section.

I f f ' is defined and finite a t x then x is a point of differentiability of J ;if f' is defined (possibly infinite) a t x , then I\-e call x a point of extended differentia- bility. 11-e let D represent the set of points of differentiabilit?. of j,and D* the points of extended differentiability, and also let S=--D,S*= -D*.

T h e first result encountered is t h a t a function nhich is rnonotonic on an interval has a finite derivative alinost every~vhere on t h a t interval. T h e conclu-

sion carries over easily to functions of bounded variation (Lc1,esgii~'. 'l'!icor.cl~i 1157: p. 223]), and to functions 11-hich are T'BG*. Intieeti, in t l ~ e s c ~cxses, not only is ?nAIT=O,b u t i n j [ S * ] = O . ([157': p. 2301, due originall>. l o T)c.lljo?-and Lusin). -1condition, phrased in ternis of a niore restrictive no ti or^ of absolute continuity, 11-hich is both necessar>- and sufficient for J to be dLitierentiab1e alniost every\\-here, under the restriction tha t , f is continuous, has heen presented 11y Pettineo [149, 1501.

T h e propert>- of smoothness (see Section 8) anti the related propert?. are of interest in investigating the diflerentiahilitl- of a function [197, 138, 1701. T h e function J is sinooth a t xo if

AZ(h )= j(.x + li) +~ ( I C- 11) - 2 j ( . ~ )= o(11) :

f satisfies condition ,I if this A,(iz) =O(i'1). ,A continuous smooth functionJ on (a, b ) has the propert). tha t the set D of its

points of tiifferentiahility has the pol\-er of the continuuili in ever>- sub-iiiterval of (a , h ) [197]. Furtherniore, ,I'' satisiies the r)arhoux contiition oil D. (See [197].) If the continuity off is replaced hy measurability, the result on difierenti- ability still holds, b u t in order to conclude the Llarboux propert\- off' we need to lrnov- tha t D is "sniall" in the sense tha t . ~ n ( D n l ) <i?z(I) for each interval IC((L, 0) [139].

If J satisiies condition ,I,then I\-e may s ta te a necessary and sufficient condi- tion for the differentiahility of /' a.e. : ,4 .~ncasiiic~l~lr /' sutis/'yi.rzg condition jltnctio?~ ,I at racl~ point oJ u .~nrus~iic~blr srf E is d$frrr?zfiahlc a.e, on E L/' und only iJjor uli?zosf ezlriy xEE fi'lerc is a?z 7, sztcll that llrl[A,(ll)]' i s s?t?n?nablc ovcr (0, 7,). T h e necessitj- is due to :\larcinliie\\-icz [102], and the sufficient>- to Stein and 2)-gniunti [169]. T h e sumnlabi1it)- of i'lrl to the exis- [A,(/?)]' a.e. is e q ~ ~ i v a l e n t tence of the L v e r i v a t i v e a.e. [169].

T h e contiition ,Iis tiispensed \\-it11 in a siniilar result due to Seugehauer [136]: A mras?tiablc fzlnction J is cqz~ivalcnt to a jlinctio?~ dilfrrrnfiablc a.c. o?z a mrusziiablr set E iJ und only if jor ali?zosf czlcry s E B tllcic is un q,> 0 S I L C ~ Itilat [A,(i1)]"i1"4(i'~-lA,(l~)) ozlci (0, r ,) , \\-here the function 4 is given is s i t ~ n ~ n a l ~ l r 11y $(x) = 1- I x ( in ( - 1, I ) , +(x) =0 else\\-here.

Properties I\-hich are relevant to a discussion of differentiahilit\- are Banach's contiitions (TI) and (T'?)and 1,usin's condition (5).For J deiined on I= la, b ] and y ~ j [ l ] , \\-e call the set { s :f(x) =y] a level set o f f . \Ye say tha t f satisfies condition (TI)if, for alnlost all ~EJ'LI],the level sets are iinite, and ( 7 . 4 if the level sets are a t niost denunlerahle. 'The contiition (S)is satisfieti if R C I and ?nPL3=0 imply tha t ? H J L R ] =Q also.

Alarchaud [100] she\\-ed t h a t if each level set of a continuous function /' is finite, then J is tiifierentiable allllost every\\-here. Iosifescu [68] has given n direct demonstration of this theorem, and has extended the result to discon- tinuous functions for 71-hich the set of points of nonnlonotonicity (i.e., points having no neighborhood on 11~11ich the function is monotonic) has measure zero. If the finiteness of the level sets is extended to denunlerability, the result is al-

DERIVATIVES 39

most totally lost, for Iosifescu has given a construction of such a function f,,for 1~11ich?n,D< E , \\-here E is any arbitrarily preassigned positive number [67].

If f is continuous and satisfies propert\- (_\-),then a theoreill of U;~I~;LCII'S [157 : p 2861 assures us tha t D has positive ineasure. 1I7eal.;ening the hypothesis b?-replacing (S)\\-it11(TI),11-e can conclude only tha t S* has an image of measure zero (a propert\- tha t characterizes continuous functions TT-hich are (TI)1157: p. 2781).Still further \\-ealcening the 115-pothesis to ( I . ? ) , \\-e call conclude only tha t D* is nolldenuinerahle, hu t can sa\- nothing ahout its ineasure.

Finally, IT-e ltnol\- tha t it is possible for a continuous function to be so badl\- behaved ns to be nol\-here differentiable. I t ilia>- he t h a t D is eiilpt!. b u t D* is not elrlpt~., as in Cellerier's exaniple 1231, or 7j.e even have D* ei11pt:-, as ill Keierstrass' function [184], \\-hich does, hon-ever, atilllit one-sided derivatives on a dense set. Even this last remnant of gooti behavior can be renioved, as in Resicovitcli's example 16, 1471, ~1-hic11 a t no point has even a unilateral deriva- tive (even infinite). These are all discussed in Jeffery 1'71 1 . I.'unctions such as Hesicovitch's are "nluch rarer" than those of 1Yeierstrass' example, in the sense t h a t the foriner collstitute a first category set i l l ~ [ a ,]11 11-hile the latter forill the complement of a first categorlT set [ 5 , 123, 1551.

11-e pass from the s t ~ c l ~ . the size of theof D and S to considerations of structure of these sets. Through use of the concept of convergence classes [62: p. 3001 one can prove tha t D is an FCa.'The same is true oi the set oi points oi left-tiifferentiabilie or points oi right-differentiability I t is not the case, 11o\~-- ever, t h a t each G6, is the set S for sotne function j.Zallorslii 1190, 1931 s h o \ ~ e d for continuous functions tha t the set S is the union of a G6 \\-it11 a null Ga, and t h a t any set of this forill is the set of points of iloildiffere~itiability for some continuous functioil. Exactl\- the same statement is true of _\-*. For a function of bountieti variation, the Ga is dropped from the theorem. Urudno [ i '? ] ex-tended the results to arbi t raq- functions, 11-it11 exact1~- the ::~~iirconditions hold- ing. Zahorski's proof has been siliiplifieti b>- Il'iranian [21 I ] .

Since the distinction bet\\-eel1 having a derivative (possibly infinite) and hav- ing a finite derivative is so often critical (see, for exainple, Section 10 on inver- sion of derivatives), it is of interest to ltnon- just here a derivative ma\-take on infinite values. 17-e ltno~\- fro111 our discussion of 1)enjoy's theorein oil Diili derivatives tha t the set { x: f'(x) is infinite ] has measure zero. Conversel~., for any set E of imeasure zero, there is a sinlple construction of a continuous, in- creasing function jE \\-it11 j h = +a 011 E (see [132 : p. 2 141). Jarnilr ['70] gave a construction of a continuous fuilctioil \\-it11 ail illfiilite derivative on ail arbi- trari1~-given ilull Gsand \\-it11 finite Zlini derivatives elsen-here, and Zahorski (1951 i~nprovecl this result to preseilt an every\\-here differentiable (in the ex- tended sense) function \\-it11 this propert!,. For other results of this nature see 13ojarslti (2001, Lipilislii (2081, 11arcus [116], and Piranian [2 101.

T h e lllost coniplete result in this direction is clue to Tzodilts [177, 198, 1'391 : For a -f inite f u n c t i o n J,necessary a n d s ~ t ~ f i c i e n t cond i t i ons .Tor tile sets El a n d E2 to be sets wheve = + a n d j'= - a respectiaely a r e : ( 1 ) El a n d E2 be F,J's wit11

??zeaszrrez e r o , aizd ( 2 ) tizere e.vist disjoiizt F,'s H I and He, wit/l E ~ C H I ,E2CHZ. Other results c o ~ ~ c e r n i ~ l g derivatives are given ill Filipczak [38, 2041, i~lf i~l i te (I;arg (491, I<rollrod [80], Landis [$'$I, A\l arcus [109, 1161, and _\l arczen-ski [ a 181.

10. Inversion of derivatives. \T'e shall be coilceriled in this sect1011 I\ it11 tha t half of the f u n d a n ~ e ~ l i a l nhicll, r o u g h l ~ tl~eorein of c , i lc~~lus , rec,~ptures,i f u ~ l c -tion from its derivative. T h e forin 1\-11icl1 this theorein u s u a l l ~ tahes ill eleiile~l- tar? calculus is. Let f be contzn~to~i>/y 1'1zc.ntd~jereiztzable on (a , h ]

the integral being talre~l ~ I Ithe sense of Ii ieina~ln. T h e requireinellt tha t f 1 he contilll~ous is us11all~- I\-ealrened i l l a course ill advanced calc\llus to the require- 11lent tha t 1' be Rie111an11 integrable. 'The example of \'oltcrra cited in Section 3 s11o11-s tha t this latter restriction ca~lllot be \\-e;~lre~led to insisting merel\- tha t ,fl be bounded. KO\\- a desirable propert\- of an integral is tha t the f ~ ~ ~ l d a n l e n t a l ecluation (*) hold for any derivative , f l , irrespective of 11-hether or not , f l is con- t i~luousor boullcled. T h e above collsicleratio~l s11o11-s tha t the Iiie111an11 integral does not have this propert\-, even for bounclecl clerivati\~es. T h e Lehesgue inte- gral does a little better. T h e relevant theorem for 1,ebesgue integrals asserts tha t (*) holds \\-henever J1 is summable. 111 particular, (*) l~olcls for 1,ebesgue integrals ~\-he~lever might fail to be ,fl is bo~lndecl. If f' is not hou~lclecl, tllell f 1

summable. The function j(.v) =.ye sin s- .~J ( 0 ) = O ;,I, af ~ ~ r ~ l i s l l e sexa~nple of clitierentiable fullction on (0, 11 11-11ose derivative is not summable over any interval colltai~lillg tile origin. T h e difficult?-, of course, lies in the fact tha t Sj" = ~ 1 3 over all!, s~1c11interval. I t is of interest to note tha t eve11 derivatives which are "tied don-11" b\. vanishing on a dense set of points (see Section 3) can 1,e so large else\\-here tha t .fl fails to be s u ~ n ~ n a b l e . llave(See [I1] and [go].) \\'e seen tha t equation (*) is not valid for Lehesgue integrals i l l general.

Perroll [148] and De~ljoj- (28, 341 illclependelltly defined integrals, both Illore general than the integral of Lebesgue's, I\-hic11 completel\- solved the prob- lem of recapturing a function from its (finite) derivative; more precisel!., of i~ltegratillg arbitrar\- derivatives so tha t (*) l~olds. ,Al tho~~gh the nlethocls of De~l joy and Perroll \\-ere elltirell. tlifferellt i l l approach, 1-Ialie [59], AAlexanclroti (1, 21, and 1,oomall (961 proved tha t these t\\-o integrals were entirel!. equiv:~-lent: i.e., if a function is integrable in one of the t11-o senses, it is integrable in the otl;er, and the t\\-o integrals are equal. Thus, this integral is usual1~- calletl tile Denj o~r-Perron integral.

111 1016 Ti l l i~l tc l~i~le [75,761 inoclified the 1 )enjoy ronstructioll to give rise to a more general integral, 1lo11- referred to as the Denjo!--1ihintclli11e integral, IT-hic11 integrated arbitrary approxiinate cleri\iati\ies of ccnti~luous fu~lctions. Descriptive clefi~litiolls of the Denjoy-I'erron and L3elljo\.-Ii11i1ltcl1ine integrals

DERIVATIVES 41

\\-ere advanced b\r Lusin [98]. For a developnlent of the Perroil integral and both the constructive and descriptive definitions of the Denjol-Perron and 1lenjo~-Ichintchine integrals, see Salrs [157]. ,An elegant development of the I'erron and the Denjoy integrals, along 11ith a proof of their equivalence, can be found in Katanson [133; Chap. 161. X detailed discussion of holy a function may be recaptured froin its derivative in a countable number of steps is given in Jeffery [7'1].

For purposes of conlparison I\-e s ta te the descriptive definitions of the Lebesgue, Denjoy-Perron, and Denjoy-Ichintchine integrals. 11-e begin 11 ith the definitions of four generalizations of the notion of absolute continuit). of a func- tion defined on an interval [a, b ] : Let I;be continuous on [a , b] and let E C [a , b ] . 'Then F is called A C(.4 C,) on E provided t h a t for any 6> 0 there exists a 6 > 0 such t h a t if ( [a , , b:,] ) is an) secluence of nonoverlapping intervals \\-it11 end- points in E and n i th E ( b , - a i ) < 6, then f(b:) -j(a:) I < E ( x u : < e , \\-here uh denotes the oscillation off on [a:, b, 1). If [a , b ] = UE:, such tha t F is .4 C ( 4 Cr) on each set EL, then F is called .4 CG(.4 CG,) on [a , b].

Descriptive definition of the Lebesgue integral: The function F is called the Lebesgue integral of a function j provided tha t :

(a) F is absolutely continuous on [a, b], and (b) F' =f a.e.

Descriptive definition of the Denjoy-Perron integral: T h e function 1; is called the Denjoy-Perron integral of a function f provided t h a t :

(a) F is ACG, on [a , b ] , and (b) F'=f a.e.

Descriptive definition of the Denjo)--1chintchine integral: T h e function F is called the Denjoy-Ichintchine integral of a function f provided:

(a) F is A CG on [a, b j, and (b) FA,=f a.e.

For a detailed develop~nent of the relevant concepts, the reader is referred to Salts [157 1.

\Ye conclude by observing tha t the derivatives considered in this section are talren to be finite. This requireillent cannot be ent i re l~. deleted, for there exist tn-o continuous functions F and G, such tha t F1=G1, yet F-G is not constant. These derivatives are equal to + co on the Cantor set , and finite else!! here. T h e difference F - G is the Cantor function. T h e first to notice the existence of t\\-o such functions was Hahn [57]. See, also, Ruzie~vicz [154], and Salrs [157: pp. 205, 2061.

11. Stationary sets and determining sets. standard theorem of elernentar)- calculus asserts tha t if the derivative of a differentiable function vanishes on an interval, then the function is constant. One might ask the question: ' ( a n 11011 large a set nlust the derivative be line\\-n to vanish, before i t is knou-n to vanish identicall)-?" This leads us to the notion of a stationaq- set for a class of func- tions.

4 2 P A P E R S I N LINLILBSIS

L ~ E F I X I T I O X . L e t e be a collection q / . / i rnc t ions de j i z ed o n [ a , 111. A s ~ l b s e tE o,f [ a , 111 w i t h tile p roper t y t ha t w l ~ e n e v e r e i s coizsfnizt o n E , tkeiz f m u s t be co i z s tn~z f o n [ a , O ] , i s sa id to be (1 s tatioiznry set jor e .

For example, if e consists of t h e continuous fullctions on [a,I ) ] , the11 the s t a - t ionary sets fo r e are the dense sets, 11-llile if (2 consists of t h e anal>-tic functions, then t h e statio11ar)- sets for e are those ~~-1l ich contai11 a t least one limit point.

If t h e collection of functions e is closed uilcler t h e operation of s u b t r a c t i o ~ ~ , then every s ta t ionary set for e is also a d e t e r m i n i n g set for e. T h a t is, tn-o mein- bers of e ~ ~ - h i c h nus st agree on all of [ a , 1 1 1 .agree on this set

111 recent years, t he s ta t ionary sets and t h e determining sets for various classes of derivatives, as T\-ell as certain related classes of functions, have heen characterized. See Roboc a n d l l a r c u s [81, Bruck~le r[ 1 2 ] ,Bruck~le ralld Leonard [ 1 6 ] , Goffman and K e ~ ~ g e b a u e r [ 5 2 ] , .\Iarcus [ 1 0 4 , 1 0 5 , 1 1 0 , 1 1 4 , 1 1 5 , 1171 , Neugebauer [ 1 3 4 ] ,S L I I ~ J - ~ SHalaguer [ 1 7 1 , 1721. T h e results of those investiga- tions n-hicll bear directly on 011s suhject are t a b ~ ~ l a t e c l in tile char t be lo\^-, ~ \ - l ~ i c l ~ lists t h e cllaracterizations of stationar). sets and deternlining sets for v a r i o ~ ~ s classes of f ~ ~ n c t i o n s on [ a , O ] . If A is a set tile11 n z l ( A ) denotes its 1,ebesgue clefi~led inner measure and card ( & 4 )i ts carcli~lality,

12. Intervals of constancy. A11 interesting functioll encounterecl b)- stuclents of a course in real variables is t he Cantor function f.This function is detined 011

[O,11 \\-it11j(O)= 0 , ] ( I ) = i t is continuous 1. I t has tile propert). t h a t a l t l l o ~ ~ g l l and nondecreasing on [ o , 11 , it is constant on ever). interval c o n t i g u o ~ ~ s to the Cantor set P. Tllus j' = ho\~-ever , fails0 except on P. I t can h e s l l o \ ~ - ~ l , t h a t f t o have a cleri \~ati \~e,finite or infinite, on a non-denulnerahle se t . X natura l cluestion to asli is \\-hetiler one call ill sonle Tva)' "smootlle" the Calltor f ~ ~ n c t i o n to arrive at a clifferentiahle f ~ ~ n c t i o l l ./ sucll t h a t . / ( 0 ) =0 , , f ( l ) = 1 , and , f l =0 on -P.T h e a~ l s~ \ -e r is in t h e negative, in vien- of tile follon-i11g result due to Zallor- ski [ 1 9 2 : p. 2 1 1 : I J n contiiziio2ts i zoncons tant f ~ t n c t i o n ./ o j l~ountdetd vnrintioiz i l c ~ s

a l inos t everyzulzere n vuizi.slziizg der ivat ive , t hen j , f a i l s to be d<fere?z t iabl t~ oiz a n Lriz- co~i7ztable set.

This theorem does not e l i~n ina te t h e possibilit). t ha t a nonconstant function be differentiable on a11 interval , J-et constant on each i n t e r ~ a l of a set of intervals \\-hose union is dense in [o , I ] . Such functions have actual1)- bee11 constructed: see Zahorslri [ 1 9 4 ] . I n fac t , t h e f o l l o ~ i n g s t a t emen t is valid [ 1 9 2 : p. 4 3 1 : A izecessnry a n d s~cgicieizt co izd i t ion t ha t E 11e the set o / zeroes o j n bolrnded tderivntive i s t ha t -E be n?z J14 set. I-sing this theore111 one can prove tile fol1o11-ing [ 1 5 ]: Le t G be a n ope^ deizse sllbset o j [ a , b ] aizd let T= NG, &4izecessury an t i s i t f i c i en t coiztditioiz t / lnt tilere be a d l f e re i z t i nb l e Jzrnction , f tdqliized o n [ a , 2 1 1 szlcil t11at , f i s constnizt 07% each conzponent in terval of G,bzlt n o t coizstnnt 07% a n y o p e n i?zterzlal coiz- t a i n i n g p o i n t s o j P,i s ilznt tile i?zter.scction o,f P zilit/l a n y a rb i t ra ry o p e n iizterval i s

. . ei ther e m p t y or h a s posztztle ineaslire.

DERIVATIVES 43

E IS A STATIONARY SET E I S A DETERII IKIKG SET

I F A;\-D OKLY I F IF A;\-D Oh-LY I F

I . Derivatives Derivatives (possibly infinite) E = [a , b ] E = [ a , b ] Derivatives (possibly infinite) of nz,( W E ) = 0 nz,( W E ) = 0

continuous functions Finite derivatives N Z ~ (-E) = 0 nz,( W E ) = 0 Bounded derivatives nz,( -E) = 0 n7,( W E ) = 0 Riemann integrable derivat~ves E is dense E is dense Bounded semicontinuous derivatives nz,( WE) = 0 m,( -E) = 0

11. A p p ~ o s i m a t e derivatives Approximate derivatives (possibly E = [a , b ] E = [a , b ]

infinite) Approximate derivatives (possibly (See S o t e Below) E = [a , b ]

infinite) of Darboux functions Approximate derivatives (possibly m,( W E ) = 0 m,( -E) = 0

infinite) of approximately con- tinuous functions

Approximate derivatives (possibly mi( -E) = 0 nzi( W E ) = 0 infinite) of continuous functions

Finite approximate derivatives in<( WE) = 0 nzi( W E ) = 0

I I I . Dini derivatives Dini derivatives of Darboux Baire E meets every perfect set E = [a , b ]

functions Dini derivatives of continuous E meets every perfect set E meets every perfect set

functions Finite Dini derivatives of continuous E meets every perfect set E meets every perfect set

functions

IV. Darbourc fzlnctions Darboux functions card ( W E ) < c E = [a , b ] Zleasurable Darboux functioils E meets every uncount- E = [a , b ]

able measurable set Darboux Baire functions E meets every perfect set E = [a , b ] Darboux Baire class 1 functions E meets every perfect set E = [a,b ] Lower semicontinuous Darboux E meets every perfect set E meets every perfect set

functions Approximately continuous lower

semicontinuous functions

.,ITote.A necessary condition for E to be a statiouary set for the class of approximate derivatives (possibly infinite) of Darboux functions is tha t n7,( -E) = O ; a sufficient condition is tha t E meet every perfect set [ 1 2 ] .

IS-e also uote that the stationary sets and the determining sets for both the class of approxi- mately derivable functions and the uniform closure of this class are the sets which are dense in the interval [ a , b ] [134] .

44 PAPEIIS I N ANALYSIS

13. Monotonicity. Accorclis~g to a theores11 of elernentar>- c,tlculus, n differ-entiable function f \\ hose clerivative is nonnegative on an interv,tl I must be ~ l o n d e c r e ~ t s i ~ ~ gon t h a t interval. This theorem llns bee11 ge~leral i~et l in m,tn>. \\ a>.s. For example, the differentinbilit> of J llas heen replaced b> ,t 11ealrer regularit>- co~lclition, the derivative h ~ s been replaced b>. v,trious t> pes of gen- e rn l i~ed derivative, and the set 011 11hich the derivative is assumed to exist, as n ell as the set on which it is assuliiecl to be ~ lo~lnega t lve , has been assumed to be less than all of the i~ltervnl I, For example, the s ta~ ldard lnonotonicit~ theorem IT l1ic11 ~tppears in the tlleory of Lehesgue i~ltegration asserts tha t a function \\ hich is ahsolutel>- conti~luous and llas a ~lonnegative derivative a e. 111ust be ~lonclecreasi~lg.A sisnilar theorem involvi~lg the ,tpproxilnnte derivative nppe,irs in connection 11it11 the integral of I>e~l jo> -Khi~ltchine.

I n this section 11e cons~cler several tl~eoresns n hose conclusions are t h a t .t

f u n c t i o ~ ~ 11-e hegin nit11 a theorem of [54] and is ~lo~lclecrensi~lg. (;oldo\\slii 'Tonelli [176] (see also [157; p. 2061).

'I'I-IEOIIERZf 011 tile ~lzterz 'nl L e t be n fu?zctlofz untlcfyilzg tile fo l lowl fzg ~o?zdi t zo izs I :

(i) f c ,zs c o f z t i ? z ~ ~ o u (ii) f f euzcts (fi?zzte or z?zjl i~zte) , e v c c j ~ t p e r h a p s o?z a dc?zl~nzcrable se t ,

(iii) f f 2 0 n.e.

T h w z f i s ?zofzdecvensifzg olz I .

11-e note tha t condition (ii) cannot be ~ ~ e a k e l ~ e d to the condition tha t the derivative exists except perh,tps on a 11~111set. This can be seen 11) consicler~ng the negative of the Cantor fu1lctio11.

I n 1939 Tolstoff [I?$] obtai~lecl an isnprovenient of the theorem of (;olclo\\- ski-'I'onelli.

THEOIIERZ.L e t f be n Jq~?zctio?z sn t i s fy i l zg tile Jollozvilzg ro?zd~t io fzs o?z n?z z?ztcvvnl I :

(i) f i s a p p r o . ~ i ? ~ z n t c l y colz t i l zuoz~s , (ii) f : , e x i s t s (.fwi f e or i?zjilzzte) except per i lnps o?z a dc?z~~nzcrnb lc se t ,

(iii) jip 2 0 n.e.

T h w z f n f zd ?zofzdecv~nsz?zg ofz I .i s c o ? z t i ? z z ~ o ~ ~ s

LAnother ge~leralization of the Goldo\\ slri-Tonelli theorem 11as o l ~ t ~ t i n e d 11) Zahorslti [192: p 191 in 19.50.

'I'EIEOIIEAI.L e t J be a fufzctzo?~ satl.>fyilzg tile fol lowifzg colzditio?zs olz a f z ilztertlnl I .

(i) f i s a D a v b o z ~ x Jzt?zctiofz, (ii) f e . ~ i s t s (j-ifzite OY iiz,fi?zite) except per i lups o?z n d e ~ z ~ ~ n z e ~ n b l e se t ,

(iii) f f 2 0 a . c

IT7e note t h a t Znhorski's Theorem is stronger than Tolstoff's in so f,tr as Zaho~slri nss~l~uecl on11 Ilarhoux continuit\ instead of approximate c o ~ l t i ~ l u i t ~ off On the other hand, his theoren1 is 11ealrer in so f,tr as conditions (ii) ,tnd (iii) ~nvolyethe orclinnr! derivative i~ls tead of the approximate derlyatiye. \Ye ould lilie a theorem n hicll implies both Tolstoff's Theorem and Zdhorslri's 'Theorem -Anol~vious candidate for s u c l n theorem is obt,tined 11) coilsideriilg the 11ealrer of the corresponding conditions of the trio theorems. T h a t is, must a I>arho~lx f u ~ l c t i o ~ l lng conditions (ii) slid (iii) of Tolstoff's l'heorem be nonclecreas- sa t l s f~ ing) 'This questioil is ansv ered in the negative, as can be seen b~ considering the example helow. 'This example is sllght modification of an exanlple fouild in [ B ~ Y: pp. 32 1, 3221

E t n m p l e . Let f be a function sa t i s f~ ing the follov i11g conditions on the inter- val (0, I ) .

(i) If ( a , b) is nn interval contiguo~ls to the Cantor qet then f(n) = O , f ( b ) = 1 ,tnd f 1s co~l t i~ luous onand n~ndecre~ ts ing [ a , b ]

(ii) If v is a t n o sided li~lli t point of the Cantor set the i l j ( r ) = 1 (iii) 13ver~ t\\ o sided limit point of the Cantor set is a point of densit! of the

set { r : f ( ~ ) = t ] . Lt is not difficult to verify t h a t this f~lnction has the recluirecl properties

So our first a t tempt to obtain s i i n u l t d ~ l e o ~ ~ s the t n o CL ge1lerali7ation of ~heorenls fails. IYllat next ' 1T.e note tha t h~ pothesis (I) of Tolstoff's Theorem implies t h a t f be in Baire cl,tss I . The same is true of 11) pothesis (ii) in Zahorslri's Theoreni. (The function in our example is in Baire class 2 , b u t not in B,tire cl,iss 1 ) \T7h,tt h,tppennf \\ e add the requireme~lt t h ~ t f he in Raire class I ? Thnt is, if f is n Darhouu function in Raire class 1 nild satisfies conditions (ii) n11d (iii) of 'Tolstoff's Theorem, nu st f be nondecreasing? 'This question \ \as aslied 11) Zahorslri [192. p. 81. I t turns out tha t this question has an affirmative ,tns\\er [201, 202, 2141, thus providiilg a tlleore~n nhich includes both the theorem of 'Tolstoff and the theorem of Zahorslri. I n fact, the folloning more general theorem is v,tlid [201, 2021

THEOREM. be f~ roper tg s u l f i c ~ c n t l y strofzg t o z m p l g L e t @ n f~ inc t zo?z - t i~eore t~c (a) d f z y D a r b o l ! x J ~ ! n c t z o f z zn Bazrc clasc. 1 W / L Z C / Lun f z~ f zec .f ~ r o f ~ e v t g6 o n a n

z?ztevval Izc. T'HG o?z I [157: p. 2211. (h) i l ~ yc o f z t z f z l ~ o l ~ s z'nrintzofz W / L Z C ~ @f z !nc t ion o f b o u ~ ~ d e d satlsjiec. j ~ r o j ~ e v f g

o n I ic. nondecveas lng o n I . T h e n a f z y D a r b o ~ ! . c R n z r e 1 t z ~ n c t z o n wi?zcir safzs$es p roper t y @ o n 1 ic. contznzl-

0117 n fzd ~zondecrcnc.l fzg o n I

'To see tha t this theore111 provides an afir1natix.e ansner to the question raised b j Z,thorslri, n e let @ be the propert>- of having, except perhaps on a denurner,tl~le set , a11 ,tpproximate deriv,ttive (finite or ~nfinite) TI hich is non- negative a.e. Condition (a) follo\\s fro111 10.8 of [157: p 2371 ,tnd co~lclition (b) is a consequence of Tolstoff's Theorenl

Roughly spealriilg, the theorem states tha t if one \\-ishes to shon- t h a t x con-ditioli is strong enough to guaraniee tha t ever). L)nrt~oux Hail-e I i i ~ n c t i o ~ isatis-f~ . ing the condition is nondecreasing, one need 0111)- s h o \ ~ t h a t ever!- corltilluous function of bounded \.ariation \\-hich satisfies the conditio11 is ~ioridecreasi~ig. (Condition (a) is likely to bc satisiied if the conclitioii is at all "reasonable.") ]:or exa~nple , one can use the theoreni to slio~v 11i;tt TolstoX's Theoren1 remains valid if approx i~nr~ tecon ti nu it^- is replaced by prepolidei-ant continuity ;ind the approximate derivative is replaced 11y the prepol~derant derivative / 207 ] .

\I7e conclude this section x i t h t \ ~ o theorems concerning IDini derivatives.

Let J' be ( L j '~~v,rfio?lc i e j i ~ e don [a,b ] wlziciz scztis$e.s (a) 1i1n S L I ~ ~ - + ~ - I , ( [ ) j'or all .YEj(()s j ( x ) 5 lini S L I ~ ~ . + ~ ~ - [a,1 1 1 , (11) D+i'zO a.e. o n [a,b ] , und (c) D+f > - C X C C ~ ~ deiz l i i?ze~( ibl~3 ~ . p o ~ s i b l y011 (1 i-et. l'i~e,v2.j i s nondecvec~siizg.

'I'his theoren1 is clue to G81 1411.()lice more, lione oi' the hypotheses of tile theoreln can be deleted \\-it11 the conclusion still valid. Other theorems oi this t!-pe, dealing n-it11 co11ti1luo11s functions, have beell advallced by (;arg [43 ] and \Yaie\\-ski [183]. 11.e state one such [43] : L e i j he n c o i ~ i i ~ i i i o i ! s , I ~ t i 7 i - t i o i ~,l '~i(t i l l ing 13c~naclz's condifioiz (I'.) (see Section 0). Lr~tQ = 1s:D-?(s) <0 . I f iizJ(Q) =0, t h e n 1 is noizdccrcnsi?ag.

14. Derivatives and Darboux functions of Baire class 1. I t was nleiitio~letl in Section I! t h a t every tlerivative belongs to Uaire class 1 and possesses the Iljar- boux propert>.. The converse is ~ ~ o t valid. 'l'hus tlie rerluirel~ielit tha t a iu11ctio11 f be a derivative is n~oi-e stringerit t l ~ a n the recli~irenient tha t j' be a I l a r l~oux Waire class I fu~lction. IIon- much more stringent? 011 the one hand, \I:ixirlloff 11201 has she\\-11 t h a t the derivatives and the i~inct ions in 1)arboux Baire c1:iss 1 are topologically equi\ralent in the sense that :rri\. Ijarboux Baire class 1 function defined on [ a ,b ] can he transt'ormed into a d e r i v ~ ~ t i v e 113. suitably t r a n s f o r m i ~ ~ g [a, D ] onto itself topologically. (See also [24, 113].) ( ) I ] the other h;~nci, the t\\-o classes of functions exhibit quite different properties. \Ye turn lo\\- to a con- sideration of sollie of these difterences.

1;or a 3 real ~lr~i l lber = 1 x: ,/'(.\-)and 1an>-,function defined on [a,b ] let Em(.[) >a \ and Ea(J )= im:J(s)< a ) .\ITe i i~ent io~led con-in Section 4 tha t Zahors1;i sidered s ~ l c h sets in tr!.ing to ch:iracterize deril-atil-es. 1 le \\-as able to sho~\- t h a t :L necessary and sufficient co~ldition t h a t J be a Darboux R < ~ i r e class 1 f~lnction is tha t each such set be an I:, set ~vitl i the propert>. tha t each point of the set be a bilateral point of conclensatio~~ the set (he called this conditioli (~)nof ?I/l).

t h e other hand, a ner-essary condition for a furiction ,f to he a derivative (possibly infinite) of a conti~iuous function is t h a t for ever>- a , Eu(J) and E a ( j )be sets satisfying tlie condition n-liich he called (A set E satisfies A/? if L;: is an F,, and every one-sided ~leighborhood of each point i11 E intersects 15 is a set of posi- ti,,, n1e;~sure.) This condition is not sufficient. Thus, each set of the t>-pe & ( j ) and Eo(j') inust be considerablg- "more dense" near its ~ i ~ e n l b e r s in order for 1'

DERIVATIVES 47

to be a derivative (possihl?- infiiiite) than iii order for j to be inere1~- a Ilarboux Uaire class 1 fl~~lctioii .Uy recjuiring even more de~isi ty of the sets E, , ( f ) and Ea(f) , Zahorski manageti to find necessary coiiditio~ls for a f u ~ i c t i o ~ ~ ato be finite derivative (co~iditioii JI , ) . 'I'he analogous necessary co~lditioii for f to be rt bo~t?ldedderivative is the still inore strillgent density co~iditioil that the sets E,(f) and EcY( / );ill be 1111 sets (see Section 4 for a defiilitio~l of J/, and sets). A sli-[ficieizt tie1isit~- co~ldi t io~ltha t the bouiitied functioii J be a derivative is that for eT7er;. a , every poilit of E,(f)(Eu(J)) be a point of (u~i i t ) density of EU('') (resp. Ea(J)) . This amounts to s a~ - ing that the fuilctioil is approxilnate1~- con-ti~luous. The converse is, of course, not true; there exist bol~ilded derivatives ~vllich are not approximately contiiluous. ITo\vever, a partial converse, 11-11ich characterizes approxi1nate1~- contiiluous functio~ls, has beell given by I>ipifislii [94] : l'ize J!l~~cfiofz /' is trppvo.vi?~int~ly iJ atzd otzlj i,fjor eeery r~ a?rd 2,colztilzlio~~s tile J Z L I Z C ~ ~ O ? ?fub( .v ) =max ( a , mi11 [ b ,J(x) ] ) is tr derivative.

We mentioned in Section 4 that the cluestioil of cizavacfevizi?rg the class of derivatives i ~ i terins of the set E, and E m has not yet been resolved. In this con- nection it should he me~lt io~ied tha t for the case of bounded derivatives 110 char-acterizatioil solely i l l t e r~nsof the structure of the individual sets E m and E, is possible. This can he seen ill the folio\\-ing I ~ J - .Zahorslii she\\-ed [192: pp. 45-47 that there are fuilctioils /, 11-11ich are not bounded derivatives, hu t such that for all a the sets Ea(,f) and E,( j ) are _114sets. Thus, if there \\-ere a condi-tion of the type de~i red , it ~vould have to be more stri~igent than the coilditioil -V4. On the other hand, on page 35 one finds the result that for every set E there exists a bou~ided derivative J and a ilu~nhera such that E =Ea(j ') . Thus the desired c o ~ i d i t i o ~ ~ calz~zot be Inore striilgeilt than J14.

So~rle ;tdditional results concer~ii~lg the sets Ea( f ) and E,(j') and _\I,sets. k = 2 , 3,4, can be found in I>ipifislii 191, 93, 951. The results of Zahorslii con- cerning the sets E,( j ) and Ea(j ' )for Darboux Eltire 1 fu~lct io~ishave been ex- tended to more general spaces by AJi6ili [129].

,Another lii~ld of comparison iilvolviilg co~iverge~it interval fuilctioils n a s advanced h ~ - Keugebauer [135]. TII this article the author gives characteriza- tions of each of the tv.0 classes prese~itlJ- under coilsideratio~i. This is done in such a \yay as to allolv ail iilterestiilg cornparis011 he t \ vee~~ classes. TTYO the t ~ v o conditio~isare stated. The first one is necessary and sufficie~it for a functioil to be a Darhoux fu~lc t io~l coilditioils together are of Baire class 1, nhereas the t ~ v o necessary and sufficient for a f u ~ i c t i o ~ ~ to be a derivative. Thus, it is precisely the second coilditioil which she\\-s how niuch lilore striilgeilt a require~nent it is for a f u ~ l c t i o ~ ~ j to he a derivative than it is for f to he a Baire class 1 Darboux fuilction. X precise formulati011 of the releveilt theorems \\-ould require liiore space than is appropriate here, so n-e omit the details.

As n-e sa\v in Section 11, a necessary and sufficieilt coilditioil that a set E be stationary for the class of derivatives is that --E have iililer measure zero [ I 171, \\-hereas a necessary and sufhcie~lt conditio~i that E be stationary for the class of Darhoux Raire class 1 fuilctioils is tha t --E be totally imperfect [as], that

48 PAPERS I N ANALYSIS

is, thrlt --E coiltail1 110 ~lollelllpt?. perfect subset. For purposes of co~ilparisoli, \ye ~lle~ltioil that every totally iniperfect set i i~ust have zero iiliier Illc,iiLir.e. 11i1i

the converse stxtemeilt is false. I t is possible, ho11-ever, for :I to~;ill!- i~nperfect set to have positive outer ~neas l~re . O ] rail I-~e decom- 111 fact, the interval [(z, posed into t\\-o no11-overlnppi~lg totall). imperfect sets [83; p. & ? I ] . i t is clear that each of these sets inust have outer measure equal to b- -a . Siniilarl?., the deternlini~ig sets for the clttss of derivatives are those 11-hosc. c-o~il~lerne~its have zero iililer Irleasure, \\-hereas the only determi~i i~lg set for the 1)arhoux Uaire class 1 fu~lctioils is the i~iterval [ a , b ] .

The renlairiirig con~parisolls i~lvolve the algehrriic ;uld topological structures of the t\\-o classes. \Ye begin by olxervi~ig that the suin of t\\-o derivatives is again a derivative. The correspoildi~ig statelneilt is not valid for the 1~)arhoux fu~lctioils of Haire class 1. Thus let j(.v) =sin (1 . v ) , ,T(C) = 1 and l e ~ g(s) =

-sill (l,/s), g ( 0 ) = 1. Then ( j '+g) (.v) =(I. ( f + g ) (0)= ,/' and g2 . .fhe f u ~ l c t i o ~ ~ s are i ~ i the required class, but their sulu is not.

011the other hand, if J is a Baire class 1 Darhoux f ~ ~ ~ i c t i o n , the11 so is ,f". This f o l l o ~ ~ s of a Haire f~~ilctioli pre-fro111 the facts tha t a conti~iuous f l ~ ~ i c t i o ~ ~ serves the Baire class and a co~itiiluous fuilctio~l of a L)arboux fun(-tion is again L3arbou.r. Rut the correspoildillg stateilie~it is not valid for derivatives. In fact if f' is a square sum~rlahle derivative, then /" is a derivative if and 0111)- if

1 J ( t ) - J(r) 1 (211 = O for all .c..

(See Iosifescu [ 6 6 ] . )S o ~ n einterestiilg related results lila>- be found i ~ iMruzka [ 2 0 51 , Iosifescu [206], Tosifescu 2nd :\ Iarcus [ 6 9 1 , Seugehal~er[ 1 3 7 ] ,Skliva~ioff [ 1 4 9 ] .IYilliosz [ I871 and \Yolff [1881 .

For ho l~~ lded aye bol~tzded fuilctio~is T\-e cttil state a sinipler result: I f f atzt-E f 2

on [ t r , 1 1 1 , f i ic~?~ are dev i~ la f i z~es co?~-hoth j~~.lzctio?rs q a ? r d ou ly i J J is a f~ f i ros i rna te l~ l ~ ~ I Z L L O L L S[ 1 8 7 ] .These results she\\- that the square of a derivative (even a bounded derivative) need not he a derivative. \Ye also see that even tho~lgh a coilti~iuous functio~iof a Darboux Baire class 1 fu~ictioli is still in that class, the correspond- ing result for derivatives is not valid. X sufhcie~lt coilditioil for such a coliiposi-tioil can be found in Chocluet [ 2 4 :p. 8 9 1 : q g i s a lo.i;lci seitzico?rti?ri~o~is dciielaficr. a11d f is a c o ~ z f i ~ z z i o ~ ~ , ~ o/ b o ~ ~ t z d c d ( - = , = ) , ti~r11t 1 1 c j ~ ~ 7 l c t i o nJ ' ~ ~ ? r c f i o n eltriitrtio?~071

f o g i s a deiicatiee. Tt is true that this theorem puts coilsiderahle restrictioils on both / and g. I t 71-ould be of interest to lino\v just 1101~much these restrictioils call be \T-eakened. To give some slight i~idicat io~l of the dif'ficulties that arise if n-e put 110restrictions (other than that of being a derivative) 011 g, 77.e state the follo\ving result found in [24; p. 891 : Ijf i s ~zo?rdecrca,si?~g a n datzd co?zti?zi~oris, szici~ t i ~ a f for r7z'r7i)1 dcriz'atiee g file / '~~tzctio?r o g is still a dciiz 'aticc, tliclz fbo~i~zdr~-E f inlist bc l i ~ z e a i .

certain other colnpariso~i has, to the best of our lalo\vledge, not yet beell resolved. Let {/,, ) be a secluence of conti~iuous fu~lctioils converging poilit\\-ise to a limit fu~lctioil/. IT-e lalo\\- tha t if each J7,is continuous and the convergence

is z ~ ~ z i f o r r n ,then f is also contiiluous. Uniform convergence is, of course. not necessav)I for the limit fuilctio~i to be conti~iuous. I t has heel1 kno\\-i1 for a long time tha t a necessary and sufficient coilditio~i for the limit function to be con- t i~iuousis tha t the convergence be quasi-u~iiform (see Hahn [56] for the relevant defi~iit io~l t ~ - p e s conver-and theorelii). One might asli for the correspo~idi~lg of gence for the class of derivatives and for the class of Darhoux Raire 1 fu~ic t io~ is : If ( J , , } is a sequence of derivatives (respectively Darboux Raire 1 fu~lctions) co~lvergiilg point\\-ise to a liniit f u n c t i o ~ ~ J, then \\-hat additio~ial restrictio~i oil the convergence is necessary and sufficieilt to guarantee tha t j also be a deriva- tive (respectively L3arboux Raire 1 function)? In this coililectioil 1T.e ~ l i e ~ i t i o ~ i t h a t uiliforin convergeilce is sufficient i l l each case, bu t not necessary. The proof for derivatives is straight for^^-ard, and the proof for L)arboux Uaire 1 functioils can be found in [14]. T h e relevant k i ~ l d of convergence for fu~ictions in Uaire class a (for fixed a ) has beell obtained h ~ - Gagaeff [$o]. The results of Oeco- nomidis [1411. 1421 are relevant to this questioil for derivatives.

\Ye coilclude \T it11 ,I precise statement of l\I,tximofi's deep theorem n hicli ~ \ ementioned a t the h e g i ~ i ~ l i ~ l g of this section. (See also r2.8: p. 901.)

THEOREM.L e t f be a , f ini te Dtrrboilx Jz[zi??ction of Ijtrire c lass 1 o n tile i~z tevcal [o, 11. T h e n there ex i s t s tr s t r ic t ly i?rcreasitzg co t z t i n~ lo i i s , f l r~zc t ioug s~iciz t ha t g(,O) =0,g(1) = 1 a?rd f o g i s a derielatiee. (See [120, 1221.)

I l a rcus [113] and 1,ipihslii 1891 consider some coilsecluences of this theoren,. 111 [121], \Iaxi~nofi shoved tha t the nard "den\ nti\ e" c a ~ i be replaced h~

the TI ords ' 'approxin~atrly co~ltinuous f u ~ l c t i o ~ i " the c o ~ i c l u s i o ~ ~ il l of the theo- rem.

111 I\-riti~ig this article the authors benefited fro111 discussioils 2nd correspoild- elices v i t h several nlathematicia~is. Particular thttillis are due to Professor Johli 011nsted for discussioils n ,he~i the project \\-as in its initial stages, and to Profes- sor Solo111o11 \Iarcus for mail>- valuable suggestio~ls concerni~lg relevant articles of n liich the ,luthors 11ere o r ig i~ ia l l~ un,ln are.

T h c firit author was supported in part 11)- K S F Grant GP 1592 'The sccond au thor was ail Y S F Cooperative Gradilatc E'cllo\\-.

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5 2 P A P E R S I N ANALYSIS

65. F.Hsiang, On differentiable functions, Bull. Amer. Math. Soc., 66 (1960) 382-383. 66. hl . Iosifescu, Conditions that the product of two derivatives be a derivative, Rev. Math.

Pures Appl., 4 (1959) 641-649. ( I n Russian.) 67. ---, Sur les fonctions continues dont les ensembles de niveau sont au plus de'nombrables,

Rev. Math. Pures Appl., 3 (1958) 439-441. 68. -, Sur un thCorkme de A. hlarchaud, Com. Acad. R . P . Romine, 6 (1956) 1169-

1171. (Romanian; Russian and French summaries.) 69. hl . Iosifescu and S. Marcus, Sur un probleme de P. Scherk, concernant la somme des

carre's de deux de'rive'es, Canad. Math. Bull., 5 (1962) 129-132. 70. V. Jarnik, Tohoku Math. J . , 37 (1933) 248. 71. R. Jeffery, The theory of functions of a real variable, I\.Iathematical Expositions No. 6,

University of Toronto Press, 1962. 72. C. Kassimatis, Functions which have generalized Riemann derivatives, Canad. J . Math.,

10 (1958)413-420. 73. A. Khintchine, Recherches sur la structure des fonctions mesurables, Fund. Math.,

9 (1927) 212-279. 74. ---, Recherches sur la structure des fonctions mesurables, Recueil Mathe'm., 31 (1924)

265-285, 377-433. (Russian.) 75. -, Sur le procede d'inte'gration de M . Denjoy, Rec. Math. Soc. Math. ;lloscou, 30

(1918) 543-557. 76. ---, Sur une extension de l'intigrale de M . Denjoy, C. R . Acad. Sci. Paris, 162 (1916)

287-291. 77. A. ICGpcke, Uber Differentierbarkeit und Xnschaulichkeit der stetigen Funktionen,

LIath. Ann., 29 (1887) 123-140. 78. ---, ~ b e reine durchaus differentierbare stetige Funktion mit Oscillationen in jedem

Intervalle, Math. Ann., 34 (1889) 161-171. 79. -, Kachtrag zu dem Aufsatze "Uber cine durchaus differentierbare stetige Funktion

mit Oscillationen in jedem Intervalle," Math, Ann., 35 (1890) 104-109. 80. A. Kronrod, Sur la structure de l'e~lsemble des points de disco~itinuite' d'une fonction

dCrivable en ses points de continuitC, Bull. Acad. Sci. URSS SCr. Math., (1939) 569-578. 81. K.Krzyzewski, S o t e on approximate derivatives, Colloq. Math., 10 (1963) 281-285. 82. LI. Kulbacka, Sur certaines propriCtCs des dCrivCes approximatives, Bull. Acad. Polon.

Sci. SCr. Sci. Math. Astronom. Phys., 12 (1964) 17-20. 83. C. Kuratowski, Topologie I , 4th ed., Monografie Matematyczne 20, LYarszawa, 1958. 84. -, Topologie 11, 3rd ed., lLIonografie hlatematyczne 21, Ii'arszawa, 1961. 85. -and \Y.Sierpidski, Les fonctions de classe 1 et les ensembles connexes putictiformes,

Fund. Math., 3 (1922) 303-313. 86. B. Kuttner, Some theorems on fractional derivatives, Proc. London &lath Soc., (3) 3

(1953) 480-497. 87. E. Landis, On the set of points of existence of an infinite derivative, Dokl. Xkad. Nauk

SSSR (N.S.), 107 (1956) 202-204. (Russian.) 88. J. Liberman, ThCorkme de Denjoy sur la dirive'e d'une fonction arbitraire par rapport B

une fonction continue, Rec. Math. (Mat . Sbornik) K.S. , 9 (1941) 221-236. 89. J . Lipidski, Mesure et derive'e, Colloq. Math., 8 (1961) 83-88. 9.0. -, Sur les dCrivCes de Pompeiu, Rev. Math. Pures Appl., 10 (1965) 447-451. 91. -, Sur certains problkmes de Choquet et de Zahorski concernant les fonction derivCes,

Fund. Math. , 44 (1957) 94-102. 92. -, Sur la discontinuite' approximative e t la de'rivCe approximative, Colloq. Math.,

10 (1963) 103-109. 93. ---, Sur les ensembles { f l ( x ) > e ] , Fund. Math., 45 (1958) 254-260. 94. ---, Sur les fonctions approximativement continues, Colloq. hlath., 5 (1958) 172-175. 95. -, Une proprie'tb des ensembles ( f ' ( z )>a ] , Fund. Math., 42 (1955) 339-342. 96. H. Looman, Uber die Perronsche Integraldefinition, Math. Ann.,93 (1925) 153-156.

DERIVATIVES 53

97. S . Lusin, Sur la rlotion de l'intL:grale, Annali Mat. Pura e t :\ppl., (3) 26 (1917) 77-129. 98. ---, Sur les propri6t6s de i1int6grale de 11. Tlenjoy, C. R . Acad. Sci. Paris, 155 (1912)

1475-1478. 99. ---, 'rrigorlotr~etric integrals and series, RIoscow, 1915. (Russiar~.\ 100. A. Marchaud, Sur une cond i t i o~~ hlath. , 20 (1933) 105-116. de quasi-rectificabilitc', 1:und. 101. J. hlarciilkiewicz, Sur les ~lollibres dCrivls, Fund. Math. , 24 (1935) 305-308. 102. --, Sur queiques inte'grales du type de Iliili, ; i n n . de la Soc. Pol, de &lath. , 17 (1938)

42-50. 103. J. 3~Iarci1lkiea.i~~ On the diillerentiability of functio~ls and sumtr~ability and :\. Zygtr~u~id,

of trigo~lotuetrical series, Fu11d. RIath., 26 (1936) 1-43. 104. S. hIarcus, Les e~isetr~bles statiorlnaires de certaines cla+ses de fonctions, C. K. Acad.

Sci. Paris, 254 (1962) 1186-1188. 105. ---, 1,es en-embles stntionnaires de certnines classe- de fonctions dhivc'ei, Atti

:\ccad, Naz. Lincei Rend. Cl. Sci. 1;ii. J I a t . \ a t . , 32 (1962) 484-487. 106. ---, 1,es fo~ictions de Pompeili, Ac;id. K.P. Tiomlne Stud. Cerc. M'it., 5 I 1954) 413-

119. (Roln:itlian; Ru->inn and French si~mmnriei.) 107. ---, Functiorli with the Darboux property and functions with connected graphs,

hIath. Ann. , 141 (1960) 311-317. 108. ---, On a theorell1 of Denjoy and on approximate deriv;itives, Mon;itih. Math .

66 (1962)435-440. 109. -Points of disco~lti~luity , and points a t which the derivative is infinite, Rev. Slath.

Pures :\ppl., 7 (1962) 309-318. (Russian.) 110. -, Retuarques sur les fonctions intCgrables au sens de liiemann, Bull. Slath. Soc.

Sci. Math. Phys. R. P . Ro~rmaine, 2 (50) (1958) 433-439. 111. -, Sur la dCrivCe approximative qllalitative, Corn. :\cad. K.P.Rotr~ine,3 (1953)

361-364. 112. -, Sur In litr~ite approxi~liative qualitative, Corn. :\cad. R.P. l io~nine , 3 (1953)

9-12. 113. ---, Sur les d6rivCes dont les z6ros forment ur1 ensetr~ble frontikre partout dense,

lielid. Circ. hIat . Palerrno, (2) 12 (1963) 1-36. 114. ---, Sur les ensetr~bles determinant de i dCrivCes iipproximatives, C. T i . :\ccrd. Sci.

Paris, 255 (1962) 1685-1687. 115. ---, Sur les eniemhlei stationnaires de fonctioni d(riv6es-finies ou iniiniei, Com.

Acad. l i . P . Romi~ie, 12 (1962) 399-402. ( l ioma~l inn;Russian and French summariei.) 116. --, Sur un problkme de Z. Zahorsiri concernant lei points 02 la d6rivi:e e-t inii~iie,

Atti Accad. S a z . Li~lcei Rend. CI. Sci. Fi,. Slat . Xat., 29 11960) 176-180. 117. --- de quaii-analyticit6, C. R.Acad. Sci. Paris, , Sur urle gCrl6ralisation de la ~ io t io~ i

254(1962)985-987. 118. E. Slarczewsl;i, Remar1;s on sets of measure zero and the derivability of tr~onotonic

functions, Prace Mat . , 1 (1955) 141-144. (Polish; English and 1iu;sian siim~naries.) 119. A. Matysiair, 0gra~iicach i pochodnych nproltsytr~atyrvnch, thesis, Lbdz, 1960. (Polish.) 120. I . Maxitr~off-', transformation of some functions into an ordinary derivative, On co~itinuous

i\nrl. Scliola So rm. Sup. Pisa., 12 (1943) 147-160. 121. ---, Sur la tra~liformation continue de fonctions, B1111. Soc. Ph\-s. hIath. I<;izati,

(3) 12 (1940) 9-41. ( l iusiian; Frerlch s~rmmary.) 122. Sur In trallsformatio~l continue de quelclues fonctioni en dfirivdes exactes, Bull.

Soc. Phys LIath. Kazan, (3) 12 (19-10) 57-81. i l i~issian;Fre~lch i ~ ~ r n i i i a r ~ - . ) 123. S. Mazlirlrie~\-icz, Sur les fo~lctions 11011-dc'rivablcs, Studia Math. , 3 (1931) 92-94. 124. ---, LIath., 2.3 (1934) 9-10. Stir les rlombrei dhrivls, F L I ~ I ~ . 125. I<. Me~iger, hIu1tiderivatives and multi-irltegrali, this I \ I o s r ~ i r . ~ , 65 (1957) 58-70. 126. ---, Rates of change and derivatives, Fund. J Ia th . , 46 (1958) 89-102. 127. I<. LIe~lger and S. Shu, Generalized derivatives and expa~isions, Proc. l a t . Acad. Sci.

U.S.A., 41 (1955) 591-595.

5 4 PAPERS I N ANALYSIS

128. S. Minetti, Sull'operazione di derivazione, Atti Acad. Naz. Lincei Rend. CI. Sci. Fis. hlat . Nat. , ( 8 ) 8 ( 1 9 5 0 ) 27-31.

129. L. MiIiSik, c b e r die Funktionen der ersten Baireschen Klasse mit der Eigenschaft von Darboux, Mat . Fyz. Casopis Sloven. Akad. Vied., 14 ( 1 9 6 4 ) 44-49.

130. A. Morse, Dini derivatives of continuous functions, Proc. Amer. Math. Soc., 5 ( 1 9 5 4 ) 126-130.

131. P . Murav'ev, A generalized derivative and its application to ordinary differential equa- tions, Izv. VysS. UEebn. Zaved. Matematika, l ( 2 6 ) ( 1 9 6 2 ) 89-100. (Russian.)

132. I. Satanson, Theory of functions of a real variable, Vol. 1 , Ungai, i iew York, 1961. 133. --, Theory of functions of a real variable, Vol. 2 , Ungar, Kew York, undated. 134. C. Neugebauer, A class of functions determined by dense sets, Arch. LIath., 12 ( 1 9 6 1 )

206-209. 135. ---, Darboux functions of Baire class 1 and derivatives, Proc. Amer. Math . Soc.,

13 ( 1 9 6 2 ) 838-843. 136. --, Differentiability almost everywhere ( to appear). 137. --, On a paper by M.Iosifescu and S. Marcus, Canad. LIath. Bull., 6 ( 1 9 6 3 ) 367-371. 138. -, Smoothness and differentiability in L,, Studia Math. , 25 ( 1 9 6 4 ) 81-91. 139. ---, Symmetric, continuous, and smooth functions, Duke Math. J . , 31 ( 1 9 6 4 ) 23-32. 140. --, X theorem on derivatives, Acta Sci. Math. (Szeged), 23 ( 1 9 6 2 ) 79-81. 141. N. Oeconomidis, Sur les nombres derives d'une suite de fonctions reelles, C. R. Acad.

Sci. Paris, 256 ( 1 9 6 3 ) 3229-3232. 142. ---, Sur les valeurs limites e t les nombres derives d'une suite de fonctions reelles,

C. R. Acad. Sci. Paris, 256 ( 1 9 6 3 ) 1208-1211. 143. H.Oliver, The exact Peano derivative, Trans. Amer. Math. Soc., 76 ( 1 9 5 4 ) 444-456. 144. A. O'iieill, Contributions to the theory of derivatives, Duke Math. J. , 12 ( 1 9 4 5 ) 89-99. 145. G. Peano, Sulla formula di Taylor, Atti della Accad. delle Scienze di Torino, 27 ( 1 8 9 1 )

40-46. (=Opere scelte, V. 1, Edizioni Cremonese, Roma, 1957, pp. 204-209.) 146. ---, Sur la dCfinition de la dCrivCe, hIathesis, ( 2 ) 2 ( 1 8 9 2 ) 12-14. ( =Opere scelte,

V . 1, Edizioni Cremonese, Rome, 1957, pp. 210-212.) 147. E . Pepper, On continuous functions without a derivative, Fund. Math. , 12 ( 1 9 2 8 )

244-263. 148. 0. Perron, Eber den Integralbegriff, S. B. Heidelberg, Xkad. Ii'iss., 16 ( 1 9 1 4 ) . 149. B. Pettineo, Quelques observations sur les fonctions dirivables presque partout, C. R.

Acad. Sci. Paris, 248 ( 1 9 5 9 ) 518-520. 150. -, Sulla derivazioile delle funzioni continue, Atti Xccad. Sci. Lett . Arti Palermo,

Parte I , 4 ( 1 9 5 6 / 7 ) 211-238. 151. D. Pompeiu, Sur les fonctions de'rivCes, Math. Ann., 63 ( 1 9 0 6 ) 326-332. 152. K . Popoff, Eber die verallgemeinerten Ableitungen, die durch ein Iteratio~lsverfahren

gebildet sind, Abh. Preuss. Xkad. \Viss. i\/Iath. Kat . Kl., 1942, No. 2 , 19 pp. 153. R. Rinehart and J. IVilson, Two types of differentiability of functions on algebras,

Rend. Circ. Mat . Palermo, ( 2 ) 11 ( 1 9 6 2 ) 204-216. 154. S. Ruziewicz, Sur les fonctions qui ont la m&me derivee e t dent la difference n'est pas

constante, Fund. Math. , 1 ( 1 9 2 0 ) 148-151. 155. S.Saks, On the functions of Besicovitch in the space of continuous functions, Fund.

Math. , 19 ( 1 9 3 2 ) 211-219. '156. ---, Sur les nombres dCrivCs des fonctions, Fund. Math. , 5 ( 1 9 2 4 ) 98-104. 157. --, Theory of the integral, h'Ionografie Matematyczne 7 , li'arszawa-Lwbw, 1937. 158. \V.Sargent, The Borel derivatives of a function, Proc. London Math. Soc., ( 2 ) 38 ( 1 9 3 5 )

180-196. 159. ii. SClivanoff, Note sur les fonctions derivCes, Bull. Math. Mech. Inst. Univ. Tomsk,

3 ( 1 9 4 6 ) 125-127. 160. U. Shukla, On points of non-symmetrical differentiability of a continuous function I.,

Ganita, 2 ( 1 9 5 1 ) 54-61.

5 5 DERIVATIVES

161. U. Shukla, O n points o f non-symmetrical differentiability o f a continuous function I I . , Ganita, 4 (1953) 139-141.

162. LV. Sierpidski, Sur les forictions de'rivCes des fonctions discontinues, Fund. Math., 3 (1922) 123-127.

163. -, Sur une propriCtC de fonctions quelconques d'une variable rCelle, Fund. Math. , 25 (1935) 1-4.

164. G.H . Sindalovskil, Congruent and asymptotic differentiability, Dokl. Akad. S a u k , SSSR , 150 (1963) 995-997. (Russian.)

165. ----, Continuity and differentiability with respect t o congruent sets, Dokl. Akad. Nauk SSSR , 134 (1960) 1305-1306. (Russian; translated in Soviet Math. Dokl., 1 (1961) 1217- 1218.)

166.-, On a generalization o f derived numbers, Izv. Akad. Nauk S S S R Ser. Mat., 24 (1960) 707-720. (Russian.)

167. A . Singh, On infinite derivatives, Fund. Rlath., 33 (1945) 106-107. 168. J . Staniszewska, Sur la classe de Baire des dCrivCes de Dini, Fund, Math., 47 (1959)

215-217. 169. E.Stein and A . Zygmund, On the differentiability o f functions, Studia Math., 23 (1964)

247-283. 170.---, Smoothness and differentiability o f functions, Ann. Univ. Sci. Budapest.

Eotvos Sect. Math. , 3-4 (1960-61) 295-307. 171. F . Sunyer Balaguer, Sobre la determinacibn de una funcibn mediante sus nfimeros

derivados, Collect. Rlath., 10 (1958) 185-194. 172. --, Sur la dCtermination d'une fonction par ses nombres dCrivCs, C . R . Acad. Sci.

Paris, 245 (1957) 1690-1692. 173. H.Thielman, Theory o f functions o f a real variable, Prentice-Hall, E~~glewood Cl i f f s ,

S. J . , 1953. 174. G . Tols to f f , Sur la dCrivCe approximative exacte, Rec. Math. ( M a t . Sbornik) N.S. ,

L (1938) 499-504. 175.-, Sur quelques propriCtCs des fonctions approximativement continues, Rec. Math.

( M a t . Sbornik) S . S . , 5 (1939) 637-645. 176. L.Tonelli, Sulle derivate esatte, Memorie della Accademia delle Scienze dell'lnstituto

Bologna, ( 8 ) 8 (1930-31) 13-15. 177. V. Tzodiks, On sets o f points where the derivative is + m or - m correspondingly,

Dokl. Akad. S a u k , S S S R ( S . S . ) , 113 (1957) 36-38. (Russian.) 178.---, On sets o f points where the derivative is finite or infinite correspondingly,

Dokl. Akad. Nauk, S S S R ( N . S . ) , 114 (1957) 1174-1176. (Russian.) 179. ---, On sets o f points where the derivative is equal t o + m or - m respectively,

Mat. Sbornik. N.S. , 43 ( 8 5 ) (1957) 429-450. (Russian.) 180. C. de la VallCe Poussin, IntCgrales de Lebesgue. Fonctions d'ensembles. Classes de Baire,

Gauthier-Villars, Paris, 1916. 181. ---, Sur l'approximation des fo~lctions d'une variable rCelle et leurs dCrivCes par les

polynBmes et les suites limitdes de Fourier, Bull. Acad. Royale de Belgique, (1908), 193-254. 182. V . Volterra, Sui principii del calcolo integrale, Giorn. di Battaglini, 19 (1881) 333-372. -183.T . Li'aiewiski, Sur une condition nCcessaire et suffisante pour qu'une fonction continue

soit monotone, Ann. Soc. Polonaise Math. , 24 (1951) 111-119. 184. K. Li'eierstrass, Zur Funktionenlehre, Rlonatsb. Akad. LViss. Berlin, August 1880,

719-743. ( I n LVerke, V. 2.) 185. C . Li'eil, On properties o f derivatives, Trans. Amer. hIath. Soc., 114 (1965) 363-376. 186. Rl. Li'eiss, On symmetric derivatives in LP,Studia Math., 24 (1964) 89-100. 187. Li'. Li'ilkosz, Some properties o f derivative functions, Fund. IkIath., 2 (1921) 145-154. 188. J . Li'olff, O n derived functions o f a real variable, Proc. Royal Acad. Amsterdam, 28

(1924) 282-285.

56 PAPERS I N -ANAL\ SIS

189. G. Young, On the derivntei of a function, Proc. London l I n t h . Soc., (2 ) 15 (1916) 360-384.

190. %. Zahoriki, Pl~nlctmengen in n-elchem eine stetige Fl~nl;tion nicht differeli~ierbar ist, liec. l l n t h . ( l l n t . Sborliilc), ( 9 ) 51 (1941) 487 510. (lius,i;ln; German iurnmar>..)

191. ---, S a r la c l ~ s e d e R,iil-e des dCl-ivce. nppt-oxim,xtives d ' ~ i n e fonction rjuelconque, .Ann. Soc. Polon. lI:~t11., 21 (1948) 306-323.

192. ----, Sur la pre~z~ikre dPrivi:e, 'l'rans. .Amer. l l n t h . Soc., 69 (1950) 1-54. 193. Sur l'ensemble de i points d e noli-dCrivabilitL' d ' l ~ n e Bull. Soc. ionciion c o n t i n ~ ~ e ,

Rlath. France, 74 (1946) 147-178. 194. --- , Uber die Tionstruletion eilier diflerelizierbarell monotolien, nicht l ionstm~ten

I'l~nlction nlit i~beral l dichter l Ienge voli l iol i i tan~interv,rI len, C. R.SociCti de Science; e t Lettres de \'nrbovie, Classe 111, 30 (1937) 202-206.

195. i iber die I lenge der Funlcte i l l \I-elchen die -4bleit~1ng unendlich ist, 'I'oh3lil1 l l n t h . J . , 48 (1941 1 321-330.

196. Z. Zalc\vasser, Sur les fonctions de Kiipcl~e, Prnce Rlat. 17iz., 35 (1927-28) .i7-99. (Polish; French suiz~rnarq-.)

197. .A. Zl,gmund, Smooth f l~nctionz, I3~1lce l l a t h . J . , 12 (1945) 47 76. 198. --, Trigono~netr ic Series, 1-01,1, I-niversitl- Press, Cambridge, 1959. 199. ---, 'l'rigonometric Series, \.ol. 2, Uliiveriity PI-ess, Cambridge, 1959.

200. B. Bojnrslii, Sur la ddrivt:e d'une fonction diicontinue, Ann. Soc. Polon. I l n t h . , 24 (1953) 190-191.

201. .A. Bruclcner, A n a f i r ~ n a t i v e answer to n problem of Zahorslci and some consecluences, l l i c h . l l a t h . J . ( t o appear).

202. --, .A theorelil on monotonicit>- and a sollltion to n problem of Znhorilci, Bull. .Ailier. l l a t h . Soc., 71 (1965) 713-716.

203. a n d C. Goffillnn, T h e boundary behavior of real f ~ ~ n c t i o n i in the upper half plane, Rev. I I a t h . Fares .Appl. ( t o nppenr).

204. F. Filipczalc, On the derivative of a discont in~~ous Colloq. RIath., 13 (1964) f ~ ~ n c t i o n , 73-79.

205. 1.. HruSIra, Une note sur les foncliol~s aux valeurs internlidinires, ensopis PCst. RIat., 71 (1946) 67-69.

206. 11. Iosifesca, On the product of two derivatives, Corn. Xcnd. I i . F.Itornine, 7 (1957) 319-32 1. (Rolzlanian.)

207. J. Leonard, Some conditions iinplq-ing the m o ~ ~ o t o n i c i t y of n real fuliction ( t o appear). 208. J. I,ipikki, Sur quelques problkmei tie S. hlarcus relatifs ii la ddrivbe d 'une fonction

monotone, I iev. RIath. Pures Appl., 8 (1963) 449-454. 209. >I. S e u b a ~ ~ e r , Funktionen, LIonatsh. I I a t h . , Uber die partiellen Derivierten ~ ~ n i t e t i g e r

38 (1931) 139--146. 210. G . Piranian, The derivative of n monotonic discontinuous function, Proc. . h e r . hlnth.

Soc., 16 (1965) 243-244. 211. ----, T h e set of nondifferentiabilit>. of n continuoui function, ( t o nppenr). 212. G. Sindalovsliii, 12ifferenti:rbility n-ith respect to congruent iets , Iav. Akad. Y a ~ ~ l i SSSlZ

Ser. l l a t h . , 29 (1965) 11-40. 213. TI'. Stepanoff, Sur les conditions d e l'exiitelice de la diffbrentielle totale, Rec. l I a t h . Soc.

l l a t h . l Ioscou (RIat. Sb.) , 32 (1925) 511-526. 214. T. S~~~i%tI ro~vs lc i , liionotonicity of fuilctions, Fund. I I a t h . ,On the conditions of 59

( t o appear) . 215. XI. \\.eisi, Total aiid partial differe~itinbilit>- in L,, S t ~ l d i a XIath., 25 (1964) 103-109. 216. H. T\.hitne>., 0 1 1 totally differentiable and smooth functioils, Pacific J , XIath., 1 (1951)

143-159.